From Nookipedia, the Animal Crossing wiki
In Animal Crossing: Wild World and Animal Crossing: City Folk, the Town Hall is the place that players can deliver letters and recycle unwanted items - as well as where Tortimer (the town mayor) can be found during certain events, usually standing outside of the structure. The Town Hall replaces the Wishing Well, Dump, and Post Office from Animal Crossing. It supersedes the Wishing Well by acting as the venue for the town's many affairs, including most, if not all, festivities. In addition, outside of and directly to the left of the structure is a bulletin board on which players can post messages and read those written by the various businesses in the town, special characters, and other villagers; at the beginning of Wild World, when driven to their new place of residence by Kapp'n in his black cab, the player is dropped off just outside the town hall - at the plaza in front of the structure.
Civic Center and Post Office
The main area of the Town Hall has a counter that divides the building in to two parts. Anywhere behind the counter is inaccessible to the player, and contains bookcases, mail sacks and the mayor's desk, where Tortimer can be found sleeping. Pelly and Phyllis also work behind the counter. There are two sections to the front desk - the Civic Center and the Post Office. Each counter has very specific functions.
The recycling bin
The recycling bin is one of the surprisingly useful things in the town, and has the same function as the dump in Animal Crossing. The player can leave any unwanted items here to be taken away at 6am every Monday and Thursday. This can be useful for any items which cannot be sold to Tom Nook. Alternatively, the player can just choose to sell them for no extra Bells.
The most useful thing the recycling bin can be used for is collecting rare pieces of furniture, or other items. Sometimes, a rare item may appear, which is likely being disposed of by another villager. These items are usually ones that do not appear in the store often, and can be taken by the player to use at no charge whatsoever. Beware though, as if these items are left past 6am on the collection days, they will disappear forever.
It is also amusing to note that if the player sends another villager an item through mail, it may appear in the recycling bin later, or another item may disappear from their house to replace said item and appear in the recycling bin. This usually happens if the item is one a villager cannot have more than one of, like wallpaper or a stereo.
The recycling bin was removed in Animal Crossing: New Leaf. If the player installs a trash bin as a public works project or places a similar piece of furniture in their house, they can dispose of trash that way. Otherwise, they will have to pay a disposal fee at Re-Tail.
Automatic Bell Dispenser
The Automatic Bell Dispenser allows players to deposit or withdraw Bells, and to pay their mortgage. The player can also ask the machine to issue them a new Shopping Card. The bell dispenser was absent in Animal Crossing: Wild World, and was moved to the Post Office in Animal Crossing: New Leaf.
The Bulletin Board
In Animal Crossing: New Leaf, the Town Hall has changed quite substantially. Isabelle replaces Pelly and Phyllis, the Bulletin Board is now outside the Train Station, the recycle bin is gone, and banking and postal services have moved to the Post Office on Main Street. The Event Plaza is also in front of the Town Tree instead of the Town Hall. However, the Town Hall is now the location of the mayoral player's office. The mayor can build public works projects and set ordinances by going to the Town Hall. The Town Hall is closed on event days, when Isabelle is in the plaza. In Animal Crossing: New Leaf - Welcome amiibo, a new CAT Machine has been added outside the Town Hall.
The music in the town hall will change depending on who is serving the player. Pelly's theme is a cheery, bouncy song with a fast whistling melody. Phyllis' contains a very droll, onbeat chord sequence with a very pitchy guitar melody. Both songs give a good idea of each clerk's personality. Smashville in the hit game Super Smash Bros Brawl has the town hall's music (along with Tom Nook's theme) play in the background. In Animal Crossing: New Leaf, Isabelle's theme played when loading the game is remixed for Town Hall.