I got my 3D printer late last week and have been having fun making a variety of models to see how it performs. I bought the Up Plus instead of one of the many kits like the Makerbot or the Reprap. The Up is more expensive, but everything I read suggested that the kit-based models require a lot more fussing to get working properly (even if you buy them preassembed) and the software is fairly painful to work with.
The Up, on the other hand, comes fully assembed and tested and has relatively user-friendly software which works out of the box. The software is a big deal, because it will automatically add support structure (for printing overhangs) and take care of other routine chores without too much tinkering from the end user.
Most hobbyist/home 3D printers work through extruding a thin filament of melted plastic--imagine a hot glue gun mated to an old-fashioned pen plotter. There are probably a dozen different basic technologies for 3D printing, but this one seems best suited for the hobby market: it is relatively inexpensive, the materials are also cheap and readily available, and safety issues are minimal.
On the downside, this method is slow, and limited in the materials you can use. The core of the unit (and most expensive component) is the print head, so hobby printers generally have a single head. That means that each model must be made from a single material, so only one color of material can be used and the support material has to be the same stuff as the model itself.
Professional units often have two or four print heads--that lets you use some other type of material for the support (making it easier to remove all the support scaffolding), and have several colors of plastic included in the same model.
I've posted some pictures of models I've build on Thingiverse. So far I've found that the printer can produce really amazing output, though sometimes the software needs some tweaking to get the best results. Support material is a pain to remove, so it's best to use the least amount of support which will still give good results.
I've also found that kids (of all ages) find the 3D printer endlessly fascinating--it's a great way to inspire interest in designing and building stuff, and my kids have already started making models in Sketchup to print.