A macro lens is capable of focusing on things that are really close.
How close? (Magification ratios explained)
A 1:1 magnification means that a lens can focus on something so close, its image on the film/sensor is the same size as the subject itself, so you can imagine that's probably about as close to the lens as the lens is long (depending on the lens design). A lens with a 1:1 magnification ratio or greater is clearly a macro lens. Just to give you an idea, this level of magnification should allow you to get close enough to a 50 cent coin so it fills the frame, and still focus on it correctly.
As with all lenses, the closer you focus, the narrower the depth of field becomes in real terms. Macro photography is often characterised by a very narrow depth of field, creating a lot of background defocus (blurred background).
Macro lenses don't only do macro
Macro lenses are designed to be able to focus close, but that doesn't mean they can't focus out to infinity as well, and they may serve as a good portrait lens too. With very few exceptions, they can focus all the way to infinity. You don't have to use them for actual macro photography (that's something I never realised when I was new to SLRs).
Properties of macro lenses in general
A macro lens of a similar quality and design will typically be more expensive, because its ability to focus so close requires a few design considerations. It may also be slightly more bulky. It may, however, have better image quality in some respects, and not just when taking macro photographs. These are generalisations only and each lens is going to be different.
Lenses for portrait photography
As mentioned above, macro lenses can sometimes be used for portrait photography too. They often exist in focal lengths that would also be attractive to portrait photographers, and are often good quality lenses with low distortion and fixed focal length.
Traditionally, portrait photographers tend to like to minimise perspective distortion ("big nose effect") so they choose longer, rather than shorter, focal lengths and stand further away. For this reason, 100mm/105mm and 135mm primes are popular focal lengths for lenses marketed as "portrait" lenses, but this won't stop you using something as wide as a 35mm or as long as a 300mm for a portrait - it's about the look you want to achieve.
This is one example of a Macro lens that should be good for portraits.