Officially, Hezbollah is simply a political organization, one of many in Lebanon. Most of its base of power is among the Shi'a Muslims of that country.
Unofficially, however, Hezbollah has a lot of power. They are in some ways like the government of some local areas where they are powerful. In such places, they serve many of the functions of government (like providing social services).
In addition, they have a very strong paramilitary wing. Many people believe that Hezbollah's military wing is strong enough that the actual government of Lebanon is afraid to take them on and, for example, try to disarm them. Because of this, Hezbollah is said to play a much larger role in Lebanon's politics than its official position would make you think that it does.