A hackathon (also hack day, hack weekend, hackfest) is a meet-up where people bring along laptops to an office space and spend the time doing some technical work to improve OpenStreetMap. This may be development of the "core" components, the editors, or any other side projects and pet projects we fancy hacking on. OpenStreetMap has development tasks sprouting from it in all directions. There's work to do in almost any programming language, as well as tasks like documentation, and even some non-technical graphics design and translation tasks.
We mostly take a fairly unstructured free-form format. People turn up and start beavering away on something, or they turn up and see what they can help with. However we can also run more structured workshops if there is demand.
In general hack weekend events are listed together with other types of OpenStreetMap events (e.g. Mapping Parties and social meet-ups). Hackathons are generally less frequent than other types of events, requiring more pre-planning/pre-announcing to get developers together.
The most canonical general list of OpenStreetMap events is Current events here on the wiki, although some local events don't find there way onto there, so be sure to figure out where you local OpenStreetMap community is actively promoting their events (and please feel free to help list events on the wiki).
Hack events have taken place in various cities around the world, but recurring events are not always to a schedule. For example London tries to have a big hack weekend at least once a year.
These german cities have got organised with recurring scheduled hackathon events:
Other tech event formats
The OpenStreetMap conference SOTM and other local conferences have quite a technology related developer-oriented theme to a lot of their presentations and workshops.
The tech community run hackathons, presentation evenings, on a wide array of tech themes, and a lot of these intersect with map technology. There's a big wide world of events out there, and it's always good if OpenStreetMap enthusiasts can get along to them and spread the word.