What are some Internet security tests?

I am tasked with come up with some examples of internet security that a web site should be written to guard against.

If my memory serves me correct.y, There are two where, if a user inputs some javascript code, in an input filed on a forward facing website, that user can inject some code where some damage can be done.

But I do not know what these sort of attacks are called. As I remember, there are more than just one way that someone can do this.