You can easily find many videos of how to tie (and use) the knot by searching online. A small note: it's Prusik (pronounced "Proo-sick") not "Prussik". If you find a video with a misspelled title, you have my permission to whup the author upside the head.
This page used to contain a description written by me, but then Mark Anderson of the vessel Riparia actually did it and wrote a much better description, so with his permission I've replaced my version with his, slightly edited. I've added a few comments and clarifications of my own at the end.
Climbing the mast is a useful skill, can sometimes be fun, sometimes necessary to repair something, and sometimes be lifesaving for you and your boat. There was a thread on going up the mast on rec.boats.cruising in the fall of 1996 that could be referred to, but here's my technique.
Remember that going up the mast is inherently dangerous. Knots could untie, lines could break, cleats could break, you could fall out of your Bos'n's chair. Then you'd fall and perhaps be killed, paralyzed, break something, or even damage your boat. Try this at your own risk. I'd suggest securing all knots tied with the ends of the line by seizing or securely taping the ends to the standing part to minimize the risk of line creep and the knots coming loose.
I rarely have anyone to hoist me up, so previously I'd used a long tackle. It's getting harder every time, and I don't like the rats nest of 150 ft. of line on deck, (particularly envisioning doing this at sea), so I was determined to try another way. Being a person who loves simplicity and frugality, I avoided buying Jumar ascenders, and succeeded, after some frustrating trials, in using Prusik knots, following a climber friend's description.
The first line I used was laid line, and smaller than the halyards. It grabbed just fine, but I couldn't slide it. Too tight. I succeeded with double braid line of the same diameter as the halyard.
First, stretch 2 halyards down the front of the mast, and secure the ends. You'll be climbing rope halyards, or the rope tails. Take 2 lengths of line, one about 10 ft., and the other about 6 ft. Take the short one and tie it to a halyard at about its midpoint with a Prusik knot. [To tie a Prusik, it's basically the same as a Lark's head, (Cow hitch) done twice. Pass a bight around the halyard, then pass the ends thru the bight, around the halyard, and back thru the bight. Lay down smoothly. See picture above -- Geoff.] Pass one of the free ends thru your bos'n's chair eye and tie the ends together with a sheet bend so that the Prusik will be at about chin level when you're sitting in the chair.
Take the longer line and tie its midpoint on the same halyard with another Prusik below the first. Tie two half hitches on the standing part to make a "slipknot" loop in both ends of this line. Adjust so that as you tighten the loops on your feet, (wear durable shoes), and stand, the Prusik will be between navel and chest height. For safety, take a third line tied to your safety harness and rolling hitch this around the second halyard.
To ascend, stand in the loops, slide up the bos'n's chair Prusik as far as possible, then sit. Pull your legs up, (the slip knots keep the loops on your feet), and slide the lower Prusik up. Stand in the loops, taking your weight off the chair, slide up its Prusik and repeat until you're up there. In the beginning you just stretch the halyards and don't go anywhere, but you will shortly. Slide up the safety harness rolling hitch also as you go. It takes some practice getting the timing right to take off the tension and do the slide, but it works. I found holding onto the halyard with one hand up high while standing in the loops, and using the other hand to slide the chair Prusik at the same time worked well. Much less tiring than hauling up with a tackle, and no line tangle. Plus, you tend to stay close to the mast.
Coming down, reverse the procedure, but I found that pulling on the Prusiks just the right way initiated a sustained slide and down I came. Watch out for rope burns.
Comments from Geoff: Mark's description needs little more. I've found that it's important to tie off the bottoms of the halyards tightly; this both helps to prevent swinging and gives you something to pull against when you are sliding the knots. As Mark implies, if the knots don't slide easily, try a different type of line. Some people suggest that it's better to make the Prusik knots using a line that is slightly smaller in diameter than the halyard.
People who have done this advise me that it's a very slow process, and that you should practice getting back down before you get up very far. One person also implied that you are going to look rather silly while climbing the first few feet.
Finally, remember that the Prusik knot is not the only one that can fail. Pay close attention to EVERYTHING that is involved in supporting your weight, or that might be asked to support you should something go wrong. And remember that if you fall, your safety line is going to take a force several times your own weight, so overspecify it, make sure it's new, and retire it if it saves you.
You are using what climbers call a classic prusik (ie the original prusik knot) but there is also the French prusik and another prusik type knot called the Kleimheist. (Geoff--It's actually the Klemheist; search online because the following URL is dead.) Go to http://www.climbing.ie/knots.html for a good diagram on how it is tied. The great thing about the Kleimheist is that it locks off as well as a classic prusik but can be easily released by loosening the loop at the bottom of the knot.
So I was think it might be worth trying a Kleimheist. DO NOT use a french prusik because although it releases easily it does so too easily and has a tendancy to slip.
Another safety tip which you may not have the luxury of on a mast is tying yourself into the rope that your are climbing. When ascending rope in a mountaineering situation we would take the loose rope below and tie a loop in this and clip it to the harness - this way you are not relying on the prusiks entirely and if the prusiks were to fail you would only fall as far as your last knot in the rope. However from what you were saying you tie off the line as the base of the mast and therefore this would not be possible.
And lastly you had mentioned that some people had advised making the prusik knot out of line smaller than the halyard. This is pretty essential because a prusik knot will not work well at all if the diameter of the knot rope is the same as the diameter of the ascending rope. (We climb rope that is 9 to 11mm in diameter and use in general 5mm cord for the prusik knot. You can use 6mm cord on 11mm rope but 6mm on 9mm rope becomes a problem because the size difference is less).
So overall I am suggesting that use of the Kleimheist with cord that is about half the diameter of your ascending rope (no less than 5mm if it is mountaineering type cord you are using) should give you a system that is secure and easy to operate. (Alternatively why not use mechanical ascenders because they are faster and easier - although I highly recommend being practised with prusik knots because you never know when you might not have the ascenders and need to improvise.)
WARNING AND DISCLAIMER. Attempt this at your own risk, and only with a proper safety line belayed by someone below. I take no responsibility for accidents or injuries resulting from the use of this information.
Gary Wilson has provided some historical information:
The knot became known as a result of the efforts of "Vertical Bill"
Cuddington who started using in the summer of 1952 to explore pits.
The Prusik knot was invented in 1931 by Dr. Karl Prusik and was
described in an Austrian mountaineering manual describing it as an
ascending knot. He died in 1961 at the age of 65, twice president
of his mountaineering club and the pioneer of 70 new mountain
climbing routes in Austria.
The knot became known as a result of the efforts of "Vertical Bill" Cuddington who started using in the summer of 1952 to explore pits.
This page maintained by Geoff Kuenning.