Answer:
Tools needed are root hook, scissors and wire cutters. Bonasi tool are recommended but other relevant stuff can be used (like an electrical wire cutter instead of a bonsai wire cutter). You will also need soil. I recommend a bag of Akadama which is suitable for most species, available on eBay. However, every specie is different. Finally copper wires (bonsai ones, eBay) needed.
Please note, I am sure you bonsai will live and feel better after following below instructions. However, it would be a great idea to spend a day at a bonsai course doing it yourself next to a professional. Again you won't kill the plant if you try but it wouldn't be 100% either. If after reading below you feel like "Oh No", feel free to take the bonsai to a bonsai shop (not a shop which also sells bonsai). They will repot it for you and show you how to do it.
I repot my bonsais at early spring.
Anyhow, first, gently check the bottom of the pot to see wires keeping the bonsai in place. These needs to be cut (and bend if necessary) to release the bonsai. As soon as wires are cut, make sure you hold the bottom of bonsai itself at all times to make sure it does not drop from the pot.
It generally need you to use root hook at the edges to free the pot. Never force bonsai out of the pot with root hook (and never pull the tree itself). Gently make your way on the sides digging vertically using root hook. Again take your time, go slow, give a break if you find yourself rushing.
Everything eventually comes out like a big soil block with the inner shape of your pot. If you are repotting at the right time, you'll see lots of tiny roots all over the place. You normally repot for two reasons. Too much root growth and soil losing its quality. Both happens in 2-3 years time generally if the tree is healthy.
When the tree is out, next task is, which is rather easier but slightly more stressful is to remove all the soil from between the roots. Holding tree from the bottom of the body with one hand and using root hook with the other. Be gentle but there are moments the you will need to pull a bit hard. Eventually what you would be looking for is pretty much no soil at all and all the roots hanging down like a long hair. No entaglements, not knots, you almost should be able to see every individual root going down from the body. There should be no visible bulks of soil or roots. It is extremely important to not to hurt surface roots (the ones near to the surface of the soil on the top). I recommend always using the root hook from the bottom, from middle to out. And maybe doing few minor work on the top if necessary.
Next is to prune the roots which is in my opinion the part which requires experience. For the purposes of this text just cut everything back to the size of the pot. If it is a late reportting, you might have to cut loads of roots and it might seem like you are doing something wrong, it is not, go back to the size of the pot. You might find yourself cutting 3/4 or 4/5 of the entire root volume, which is normal. Also, you would be looking into removing unusually tick root formations but skip this part because you might hurt good ones by mistake.
When all the roots cut back, I generally spray the roots with water before putting it a side to prepare the pot. Keeps them humid.
Back to the pot, clean well with water. Check plastic grids on the bottom holes as these can come with the soil. If they are gone, you should put another one (or a similar stuff) to stop soil going through the holes but let water go.
Next soil. If you would like to do everything properly, I recommend using a sieve to separate the Akadama to 3 sizes, say fine, medium, coarse. If you can not do this step skip it, your bonsai will still feel better.
Coarse and followed by some medium needs to go to bottom, then the tree goes in. Say an amount of soil in the bottom, tree in the middle same amount on the top. Anyhow, after coarse, some medium and tree, use new bonsai wires to fix the tree into the pot. A bit difficult to explain in writing, just use two diagonal wires coming from the holes at the bottom, over the roots, by the body back to the holes. Connect them at the top, turn few times with a pliers. Hope this made sense. Tree should not move in the pot at all.
Then more medium soil, lots of shakes and pokes to make sure soil goes all over the roots (without hurting them). And finally fine soil on top. No roots should be visible, but don't get overly worries for one single top root sticking out. Push it in (or cut it). More shakes.
Watering. Immediately. You have a watering can or a hose with a good sprinkler don't you. In any case idea is to water without messing out the soil (it will get harder in time but you can water half out if you put it under tap first time). Water should come out from the bottom immediately, idea of fine/medium/coarse grain ordering is to make sure no water retained other than the ones in the Akadama granules itself. We want water to just drain immediately.
Keep watering until it runs clear from the bottom (will be mouldy at the beginning).
Job done. Do not feed for about 3 weeks to let the roots heal. You feed your bonsai don't you. (search biogold if necessary). Then start feeding.
Hope this helps. Again, I recommend a course if you are serious but above should be sufficient if you have this small tree given as a Christmas gift.