In the game there are various skills or professions. There are some standard ones anyone can do look cooking, fishing, first-aid, and now archeology. There are two you can choose from mining, skinning, engineering, and various others (some gathering, like mining, and some production, like engineering, and they often go together in obvious pairs).
Most of them work in a similar way - the skill has a basic level from 1 to around 525, and these are in stages. You start with ability to go to 75, but at 50 you can buy more training to go up to 150. The things you do are only possible at certain skills. Anyone can mine copper, but mining tin is a higher level and then silver, gold, and so on. So you have to work your way through the skills doing the things you can do at that level. What is cunning is trying to do low level things does not get you more skill points, or gets them very very slowly. So you cannot up your mining by just mining copper - you will have to go to nastier parts of the game to find rarer minerals and metals once your skill level reaches a certain point. Well, you can still mine and process copper at higher levels, and so make stuff, but they don't get more skill points - that is the point.
With most skills you also get to learn various spells or recipes as you go - so you learn things you can make - like bronze from tin and copper, and so on. You have to reach certain skill levels to learn these, but then doing them gets you up the skill levels faster. In general, doing the thing you have just been able to learn gets you 1 skill point each time, but several skill points later it changes colour and starts needing two or three or more operations to get each skill point. Eventually you are far enough ahead you get no skill points for the operation, but by then you can learn a new operation that gets you skill points faster. The operations are colour coded red, yellow, green, and grey as you progress and they are worth less.
You work up a ladder of skill points and things you can do. And in the process you make useful things...
Fishing is, however, rather different, and I have a rough idea how it works. It has changed over time from what I can see on the forums, but this is what it looks like to me now.
1. You get skill points for fishing, but it does not matter what you fish up - the rate of skill points is the same regardless and slows down as you progress. Up to level 100 you get one point per thing you fish up. Then you need more than 1 per skill point. Eventually you need dozens per point. The actual process is casting the line, waiting up to around 20 seconds for the float to bob and then right clicking it. Rinse, repeat, drink lots to avoid boredom. You also need to equip a fishing pole first and put your knife/wand/gun back afterwards - tryign to fight mosters with a fishing pole is not good!
2. There appear to be different areas to fish and they work at different skill levels it seems. So when you start you fish up crap lots of the time (cloth, tangled fishing lines, all sorts). Your relative skill level to the area seems to be the factor. If you try and fish too high an area you only get crap. However, unlike mining you can keep fishing a starting area and get skill points at the same rate. Just that as your skill goes up you get the fish every time rather than crap. The fish are better from the higher skill areas though.
3. The fish have some uses, usually better cooked (another skill you can learn). The crap is usually sell-able. Some of the fish are really useful to improve heath and add extra strength or some such. Some have magic properties (turn you in to a pirate for an hour) and so are highly sell-able.
4. Unlike other skills there are no extra things to learn - just the levels. Being a higher level allows you to fish more interesting fish with higher chance. That seems to be all. You can boost your level with special rods, and shiny things on the line, which increase chances of fish not crap on the line. Someone pointed out in some areas the crap sells for than the fish so this is not an advantage in all cases.
Some quests involve fish, and some achievements involve fish in odd ways.
But basically it is a skill to accumulate when you are really bored and waiting for something or someone.