Is there anything that a potential employer look for in any portfolio, or is it mostly job-specific?
I would include implementations of various algorithms and data structures in the language that the company is interested in, as well as work in a variety of languages. However, it's important to show good code, and not just code.
One of the good things to do when you're starting out is to get involved in an open source project. You can get some actual coding experience and be able to demonstrate it with code that does something of value.
I like to see passion about something involving technology or science.
It doesn't matter if it's unrelated to the company or technology used the ability to be passionate about something dead and buried commercially such as writing games for the Atari 2600 shows the right sort of characteristics.
In fact it's far better to show interest in something leftfield and obsolete as they are less likely to want to involve their pet technology in projects where it isn't even in the top 10 best technologies to use ;)
I think few interviewers care what your code sample contains, as long is it's well-commented, efficient, and bug-free. It's probably a good idea to spend a few hours polishing up your code samples before the interview. Make them good, but not perfect -- you don't want it to look like you copied them out of a book or something.
Try to make your samples relevant to the job you're applying for, as that speaks to your experience with the kind of work they do. If you're interviewing for a web development position, bring code from that MVC framework you've been designing. If it's a game programming position, bring DirectX code.
Personally, I like seeing library code, especially if the person I'm interviewing designed its API. That gives me an idea of how they write code other people have to use and maintain.
My #1 would be that the code somehow relates to the problems that I, as the hiring manager, have to solve.
Beyond that, I'm less interested in what problem is solved by the code than by the code not being a problem itself. It should look beautiful, read well and be appropriately commented.
More than just code, show final polished apps as that kind of thing especially looks valuable to a leader. Especially show something that is currently in heavy use like a major web app or something like that.