Here's a good primer article:
In reality, a securities trader could have many different functions including, but not limited to:
-executing trades for clients
-buy and selling securities for his/her own money or the firms money
-executing trades for the sake of hedging (limiting) risk to other traders portfolios
-researching some trade ideas
-servicing trades for clients (for example replacing bonds that are up for maturity with new bonds that are expiring at a later date)
-analyzing and issuing reports on specific entities that he/she will trade
-working with other banks to facilitate trades in other securities
-issue profit and loss statements based on the overall trades he/she has made
-buy and selling for the sake of making risk-free profits (called arbitrage trading)
...and really the list goes on and on and on and on. It really depends on what the specific job title is, but that is some of the generic descriptions that you'll run into.
I will add, that in order to become a securities trader, an additional requirement I've seen on some job descriptions is that they have data manipulation/programming skills (like VBA, C++, SQL..etc) because you'll need to be able to analyze lots of quantitative data. Of course, a very strong math background will come in handy. I've seen investment banks really start to recruit heavily from Mechanical engineers and such who have those skills, but really to work for an investment bank you need to do well (with a GPA above 3.5) and demonstrate that you have both the analytical, quantitative, qualitative, and stamina to handle a fast-paced environment.