Any of a number of things: maintaining voter lists, conducting outreach, local fundraising, talking to the media, attending workshops conducted by the national or state party, planning events (and supervising the volunteers who help with all those things).

One example: The Republican Party county chair in my former home county was in the paper every other day, staking out a Republican position on almost every issue with admirable consistency.  (He was also, I should add, a genuinely good person and not a Rovian sleazeball.)  I saw him or subordinates at almost every major community function I attended.  He and the Republican Party were everywhere in the county.

Now, different states may have different methods of organization (relying more on the state leadership, for example), but the principle is the same: visibility as Democrats, putting our principles into local circulation, and keeping tabs on every possible opportunity to bring another voter out to vote for Democrats.