When it comes to ease of blowing and the production of a good tone, the quality of the mouthpiece on a sax or clarinet is pivotal.
'Pivotal' isn't an idle choice of word either - at a very simple level there are two things you need when playing an instrument; your physical technique and a working instrument...and the mouthpiece sits between the two.
The importance of the mouthpiece is also often expressed by the idea that the tip of the instrument's bell is the least important part of the equation, and that as you move closer to the mouth the more your equipment is going to have an effect on your playing.
Either way, it's a done deal that a poor quality mouthpiece will hinder your progress and a good one will allow you to concentrate on building a good embouchure - and thus a good technique and tone.

Beginners often find this hard to understand - or, perhaps more accurately, parents of beginners don't often appreciate how hard it is for their children to persevere with a mediocre mouthpiece.
It's very easy to demonstrate, and I must have done it hundreds of times. When I've finished working on a student sax or clarinet that comes with a cheap mouthpiece and the client calls in to collect it - I play it using their mouthpiece, and then play it again using my own.
The difference is always clearly audible, even when a simple three or four note phrase played slowly is used. The tone is fuller, the upper harmonics are less shrill, the midrange is cleaner and the bottom end is rich and sonorous.
OK, so a significant part of that tone is down to my own technique - but by playing on both mouthpieces it's easy for the listener hear how much influence they have on the tone. What can't be heard, and is just as important, is how easily the better mouthpiece blows.

I've been present many times when a student player has swapped a cheap mouthpiece for a decent one, and they usually say one of two things after trying out the new 'piece' - it's either "Wow! It's so much easier to play!" or "It sounds like a completely different instrument!".
If you're still sceptical that a mouthpiece can make that much difference then I can't help you any further - and either you or your child is doomed to forever wrestle with a mouthpiece that's been made with little or no thought as to whether it actually works or not.
If, however, you're curious to find out more - read on.

What distinguishes a good mouthpiece from a bad one is the accuracy with which it's built, and there are a lot of factors to take into account.
Certain parts of the mouthpiece must be absolutely flat or level, otherwise it will be very hard to blow - and the slightest error in internal dimensions can result in a very brittle and unpleasant sound or a stuffy tone.
Cheap mouthpieces are often moulded out of plastic, and very little thought is given to whether one piece is the same as the next piece to roll out of the machine. What you get is what you get - it's a lottery in other words, and the odds are stacked against you.
A better quality basic mouthpiece may also be made out of plastic - but it will be a harder plastic, and will be made on a machine that faithfully reproduces the desired specifications time after time. More expensive mouthpieces will be made out of a hard rubber compound - or even metal - and will be carefully machined to size. There may also be some hand-finishing involved.
For such a seemingly insignificant part of the instrument, the design of the mouthpiece is really quite complicated - which is why technicians ( often known as 'refacers' ) who are able to work on and change the tone and response of a mouthpiece are held in very high esteem by experienced players.
You don't really need to know the ins and outs of mouthpiece design at this point, but if you're interested I thoroughly recommend having a look at Theo Wanne's excellent 'mouthpiece primer' at www.theowanne.com/mouthpieces101/glossary.php.

From the player's perspective a poor mouthpiece will force them to focus on simply getting it to produce a note. What comes out might be stuffy or squawky ( like a duck! ), but as a beginner they'll have no 'point of reference' and won't realise that as much as half their efforts are being wasted on working around the mouthpiece's inadequacies. That might sound like a made up figure, but a decent mouthpiece can quite literally make the instrument play twice as well as with a poor one.
And, if I'm being harsh ( but honest ), beginners seldom have any idea of what a good tone is. As long as it sounds vaguely like a saxophone or a clarinet, it'll do. And that's fair enough - whenever any of us embark on learning a new skill we can't be expected to know everything all at once - but nothing ensures success like decent equipment that's of a given quality.

By now many of you might be wondering whether the mouthpiece on your sax or clarinet is a good one or not, so the first thing to do is look for any markings on it.
Cheap plastic mouthpieces are likely to be unbranded - they may sometimes have a letter or a number stamped on them ( such as B21, for example ) but there won't be a maker's name or logo. Very often the mouthpiece will be the one that came packaged with the instrument.
If your mouthpiece fits this description then you almost certainly have a poor quality mouthpiece.
If it has a name on it, do a search on the internet - see if you can find anyone selling them. If you can, and the price is suitably high, you may already have a decent mouthpiece...though that doesn't always means it's the right one for you.

So what are the options?
A quick peek at a website where mouthpieces are sold will probably send a chill down your spine. Prices running into three figures aren't uncommon for mouthpieces - and the choice is likely to be vast, and confusing.
Fortunately there's a very simple solution. For sax players, get a Yamaha mouthpiece - for clarinet players, either a Yamaha or a Buffet mouthpiece. The price for these pieces is around the £30 ( $25 ) mark.
The reason for this simplistic approach is, well, simple.
Yamaha and Buffet have been making quality student instruments for decades and have invested considerable time and money in the production of a basic, good quality mouthpiece that ensures that players who purchase their instruments will have the very best chance of getting a decent tone out of them ( you might have realised by now that if you own an instrument made by one of these two companies, you probably already have a decent mouthpiece...but it's worth checking in case you bought it used and the previous owner replaced the mouthpiece ).
In short, they have become the 'industry standard' for student mouthpieces.
Note though that Yamaha make two ranges of mouthpieces - a Standard series and a Custom series. You want the Standard series ( made in hard plastic ).

There is still a choice to be made, however, and that's in the model of mouthpiece you buy.
Mouthpiece models are often described by a combination of letters, numbers and symbols. For example Otto Link mouthpieces use a number and, sometimes, a symbol - such as 5*, with the * symbolising the word 'star'. Yamaha use a number and a letter - 4C, for example. These markings indicate the characteristics of the mouthpiece - which is another topic in itself!
As far as the beginner looking to buy a Yamaha piece is concerned, the basic range runs from a 3C piece up to a 7C - and the smaller the number the brighter the mouthpiece will sound ( the higher the number, the darker ).
Brightness and darkness are terms that describe the general tonal response of a mouthpiece, and in very simple terms you can think of brightness as being edgy and piercing and darkness as being full and mellow.
It might sound like brightness isn't going to be something you'd want, but a bright mouthpiece is often easier for a beginner to play. Conversely, a dark mouthpiece can turn out to be a bit stiffer to blow.
In recent years there has been a trend towards darker mouthpieces, and I often see students struggling with such pieces. As such I would recommend going for a Yamaha 3C or 4C mouthpiece - perhaps a 5C if you're adamant that you want a warm tone. Leave the other models to players who are able to try the pieces out and make their own minds up.
Clarinettists also have the option of buying the basic Buffet mouthpiece. It's a bit more expensive than the Yamaha, and my own personal preference would be for the Yamaha piece - but you won't go far wrong with the Buffet.

There are other brands available - for example the David Hite range of basic mouthpieces is often recommended as a decent alternative - but one thing to watch for is the 'shop own brand'. These might well be good, and can often be a bit cheaper than big brand pieces, but they won't be as well known and therefore as well documented when it comes to how well they work. If you can call upon the services of an experienced player who can try them out for you, they might represent a bargain - but otherwise you're better off playing it safe.
Beware too of 'the push'. You'll see that it's possible to spend a lot of money on a mouthpiece, and there's a huge range that come in at about two to three times the price of student pieces. Names like Selmer, Vandoren, Rousseau and Otto Link dominate this section of the market.
Such mouthpieces are of good quality, and you may well be told by a helpful sales assistant that these pieces are far better than the one you're looking to purchase.
They're right - but until you have the playing skills to test such pieces and choose the one that gives you the sound and feel you want, you're better off sticking with a basic piece ( a year is sensible amount of time to practise before considering a 'pro quality' mouthpiece ).

You may also be offered a fancy ligature ( the clamp that holds the reed in place ). At this stage you can forget it - there are endless debates among experienced players about whether such ligatures offer any benefit, and if the professional players can't agree then you certainly won't benefit from any supposed difference.
If you have a cheap mouthpiece you probably have a cheap ligature. It will do - and will probably fit your new mouthpiece - but a good quality basic ligature is easier to use and will hold the reed on with more accuracy. If the ligature you're offered costs around half the price of the mouthpiece, tell the seller to think again.
Many of the finest players I've ever heard use basic ligatures - buy a more expensive one when your playing skills are good enough to allow you to hear any difference ( or not, as the case may be ). I have used a variety of ligatures over the years, I now use plain metal ones.
You'll also need a mouthpiece cap, which protects the vulnerable tip of the mouthpiece and the reed. Again, you may already have one - and it will probably be more than adequate. If offered an expensive model, just ask for a plain one - it'll do just fine.
You may also be offered mouthpiece patches. These are small stick-on cushions that are placed on the top of the mouthpiece. They help to prevent irritating vibrations coming up through the top teeth - and they also serve to protect the mouthpiece from being marked by the teeth.
Unless you're having very real problems dealing with the vibrations ( you do get used to them in a very short space of time ) you really don't need them - and to some extent they dampen the amount of feel you get from the mouthpiece. They can be useful for the first few weeks of playing, when a beginner is inclined to bite hard on the mouthpiece - but it's best to stop using them as soon as possible.
Having said that, I carry a few in my instrument cases in the event I get what I call 'itchy teeth' - those odd occassions where I experience that 'fingernails down a blackboard' sensation. It happens about once a year or so.

The only possible problem you might run in to if you're very unlucky is that the mouthpiece is either too tight or too loose a fit.
In the case of a tight mouthpiece you'll probably find that it just needs a bit of cork grease. At worst you'll need to have the cork sanded down a little. If it's too loose then you will need to have the cork replaced ( either on the mouthpiece or the crook/neck ). This isn't an expensive job.

If you're a player struggling to get a good tone without having to blow your lungs out, you're probably convinced of the benefits of having a decent mouthpiece by now - but if you're a parent of a child who's learning the sax or clarinet you might still be wondering if it isn't yet another unnecessary expense.
To you I would say this. If someone offered you a gadget for £30 that was proven to double the number of miles per gallon your car gave you, you'd buy it in half an instant wouldn't you? This is effectively the difference a reliable mouthpiece makes to the player - it's the cheapest, most effective upgrade you can make.