I think the word "trader" means a lot of different things to different people, so while you're asking what you think is a straightforward question you'll get a lot of different answers because the people answering are all thinking of a different position.
For example, to be specific hedge funds are one entity that make a number of directional bets on markets. You enter the hedge fund business as an analyst out of undergrad, generally with a very technical degree like match, stats, electrical engineering... To go direct to a buy side job like this you likely needed to graduate at the top of your class from a top 10 engineering/math school, i.e. MIT, Princeton, Cal, Stanford... Even then the acceptance rate is very low. Alternately you can go to a sell-side job at an investment bank, a slightly less desirable job but requiring basically the same prerequisites. After 3 or 4 years there you try to move over the the buy side. Finally, you can get an MBA from one of 5 or 6 schools (Stanford, Harvard, Wharton, Booth, maybe Columbia and a couple of others) after getting the technical undergrad and 5 years or so of work experience, get a buy-side internship between the two years, and get hired as an associate at a hedge fund if you didn't have any previous buy-side experience. Again uber competitive to get the job in addition to what you had to do to be accepted to the school and get the internship. In all those cases, you're not given a lump of capital and told to go forth and do good things. The fund will have an investment thesis, you will research and make recommendations based on this thesis. If it's an HFT firm it will be slightly different as you'll be designing algos, but again based on the thesis of the firm. An investment committee will look at your recommendations and use them or not. Eventually you'll be on this committee.
In all cases base comp is high but the price you pay is high as well, expect 80 hour weeks doing a bunch of crappy work with much of your comp tied up in a bonus earned for the previous year but not paid until the following June or July and only if you're still there then. A very high proportion of the people who go this route come to the realization somewhere along the way that money isn't nearly as important as they thought it was and move on to something else.