No photo editing software can really let you remove people from the picture (at least not without leaving a blank space behind them). What they can do is allow you to cover over the people. This is usually done with some form of "cloning" tool. The cloning tool makes it easy to copy one part of the picture and past it over another.
As an example, let's say that you took a picture at a beach and someone left a bucket in the sand that you want to get rid of. You could clone parts of the sand that look similar to the sand around the bucket and copy them over the bucket.
For some photos, cloning is easy and produces great results. For others, it can be very difficult. It works really well when you can past over the offending elements with parts that are either uniform (like sky, a solid colored wall) or extremely random (like sand, leaves, grass). It doesn't work well when you need to clone in details.
With that said, most photo editing packages provide some cloning capabilities. In know that Photoshop does. I suspect that GIMP and Picasa do as well.
Photoshop has tried to improve the basic cloning tool with the "healing brush". This tool goes beyond simply pasting in other parts of the photo and tries to match tectures, colors, and such so that the pasted in portion blends more naturally. It tends to work really well in some cases (like eliminating blemishes on skin) and not so well on others.