Matchstick models are scale models made from matches. Regular matches are usually not used, however, but a special modeling type which do not have the combustible heads, and can be bought from art and craft shops. Though before the production of these, actual matches were used with heads trimmed off, or kept on to add colored detail.
Originally, matchstick models were a pastime of prisoners (especially naval prisoners of war) during the 18th century. At the time, better funded modelers preferred to use more replicated parts for their models, like professionals today, and the poor couldn't afford to use up so many matches.
The matches are cut by means of a sharp knife and fixed together using glue, often being held in place by paperboard "formers" until the glue is dry. While the smallest gaps can be filled with glue, larger ones can be filled with specially carved matches. While a number of hobbyists prefer to build their models from scratch, many kits are available, consisting of instructions, pre-cut card formers and sufficient modeling matches for the project.
Checkout this story on a retired carpenter from Great Britain who has built famous buildings and used over a million matchsticks to build versions of famous buildings around the world....
...or one of the models of this guy from Iowa.
In addition to kits you can also buy books and even license plate holders and T-shirts on matchstick modeling. Many craft stores sell everything you need to get started, too. Some links: Getting Started, Building Models, How to Create Matchstick Models, Matchstick Marvels and Beginners Guide.