Closer to a distant subject, Paul, or closer to a nearby subject?
For distant subjects you need a telephoto lens, which is one with a longer focal length. For example the Sigma and Tamron 70-300 lenses have a focal length of 300mm when fully zoomed, which is over 5 times the reach of the kit lens with its maximum of 55mm. So distant objects will appear over 5 times bigger with a 70-300 than they will with an 18-55. The lenses mentioned above are good budget choices; lenses with focal lengths of up to 300mm are available quite cheaply, but going longer than 300mm is pretty much always going to be expensive.
But to get closer to nearby subjects you need a macro lens, which is one that's designed to focus on really close things. A true macro lens has a magnification of 1:1. This means that you can fill the frame with a subject the same size as the sensor, which is roughly the size of a postage stamp (22x15mm). True macro lenses tend to be a bit pricey though. Many general-purpose lenses claim to have a "macro mode" which simply means that they can focus fairly close; for example the Tamron 70-300 mentioned above can achieve 1:2, so you can fill the frame with an object twice the size of the sensor or 44x30mm. In general if a lens says it can achieve a magnification of 1:N then it can fill the frame with an object N times the size of the sensor. Unfortunately you usually can't tell the magnification from the name of the lens, so you need to read the specifications or reviews, or ask.
Does that help?