OS X's memory management system uses lots of virtual memory. Since virtual memory is actually space on your startup drive, having a startup drive that is almost full is sure to degrade your Mac's performance.
There are other concerns related to a nearly full startup drive, besides the obvious lack of space for new applications or documents. Almost all applications use some type of temporary file while you're running them. If there isn't enough disk space to create the file, an application may crash or fail to start up.
I recommend keeping 15% free space on your startup disk; more is better, of course.
You can find out how much free space you have available on your startup drive by right-clicking the startup drive's icon and selecting 'Get Info' from the pop-up menu.
The Get Info window displays the drive's total capacity, available space, and used space. If the available space is less than 15% of the listed capacity, then it's time to consider freeing up some disk space, or possibly replacing the startup drive with a larger model.
You can find suggestions for how to free up disk space in the How Much Free Drive Space Do I Need? article.