Not very much. The British captured Detroit in 1812 shortly after the start of the war, and Fort Niagara at the end of 1813. However, they didn't hold either position for particularly long. 1814 saw a half-hearted invasion of northern New York and some fighting on Lake Champlain. That's about it, unless you count the Burning of Washington and the Chesapeake campaign. Even that was little more than a very successful raid.

There are several reasons for this. First and foremost, Britain had bigger problems to worry about (in the form of Napoleon) than a frontier war in the backwoods of America. The British also remembered the lessons of the Revolutionary War: America was too big and two decentralized for any number of land victories to mean much. The British very likely could have taken and held New York City again if they wanted to, but doing so had not gained them in the last war. Rather than attacking from the sea, the British chose to operate out of Canada, and focused mainly on keeping the Americans off Canadian soil. Fighting in the interior of the US - Ohio, and along the Mississippi corridor - was left to Britain's Native American allies under Techumseh.

One of the more entertaining things that came of the War of 1812 was Dalhousie University.

The British marauded across Maine quite successfully, and occupied the (still existing) city of Castine, Maine. They taxed trade and build up a fund of many thousands of pounds. At the war's end, the British left with this money and took it to Halifax, Nova Scotia.

A few years after the war, Lord Dalhousie of Halifax proposed that a scottish-style university be founded in Halifax using the funds. Dalhousie University was founded on plundered money.

Dalhousie University later danced with insolvency in the late 1800s and was rescued by a publisher of pulp fiction novels, George Munro. As a result Munro Day is now a favorite day for Dalhousie students to enthusiastically salute the generosity of the savior of the University.