Completing community buildings increases your town's population cap and allows you to build more houses.
Once the construction time of a building has elapsed, the building will be wrapped up with balloons if you have the required materials in your barn to finish construction, at which point you may choose spend those materials to complete it. If you are missing materials, the building will be surrounded with wood scaffolding with a yellow construction helmet icon floating above. To see what materials you'll need to finish each building, tap on the construction helmet.
Some buildings have animated features (just like factories when they are working). They can be simple like the Post Office man checking his watch, while others are more complex like the Roller Coaster or Fashion House. Visit Ernie or higher level friends to see which buildings will animate in your town, or refer to the table below.
NOTE: If you build more than 1 community building at a time it may throw off the train algorithm and bring you extra unwanted materials. You can read more about correcting this here.
TIP: Complete the community building with the lowest amount of materials required first, then continue in that order until you are back on track.
From level 60, new Community Buildings require special tools that cannot be obtained from trains. These tools can only be produced by a special building, called Tool Exchange. The Tool Exchange produces jackhammers, drills and electric saws.
Currently there are 72 community buildings at max population: 22,775
Coastguard, Hydroelectric Power Plant, Lighthouse and Shipyard buildings must be placed on water.
General information about tables can be found here.
Community Buildings with stars (★) are animated.
G, S and B stands for Glass, Slabs and brick Respectively
A spreadsheet of materials needed before the March 2019 update has been created here. http://bit.ly/TSLegacyMaterials Only use these if you play on Windows, Bluestacks, or NOX and have not received the update yet.