Town Hall
This building's level determines the max level of all your other buildings. So, if you want Barracks level 6, your Town Hall has to be at least level 6. Also, increases gold output and gives a good amount of fame when upgraded.
Barracks
Troops are built here. Every level of the Barracks will unlock one unit, but there are typically additional requirements to meet as well. The building doesn't give any special bonuses when upgraded.
Hero Tower
Using the Hero Tower, you can appoint a Castellan. The Castellan will defend you if attacked and their INT will increase your gold output. In addition, the Castellan gains good experience points when you building things, train troops, etc in the castle. Upgrading the Hero Tower gives your Heroes +1% exp per tower level.
Tavern
The Tavern is where you can recruit heroes. In order to upgrade it, the Hero Tower has to be a high level than it. At certain levels it will display more Heroes that you can hire: Level 2 = 3, Level 4 = 4, (?), Level 9 = 7, Level 10 = 8. Note that a higher level tavern will have a higher chance that any heroes within it are a better quality. A level 1 or 2 tavern will almost always have white colored heroes with white stats. A level 10 tavern has most heroes blue colored, often with multiple colored stats. Level 10 is the only tavern level in which purple heroes have been seen. Also, the level 10 Tavern only requires a level 8 hero tower, which requires a level 7 Town Hall.
A tavern will normally refresh once per hour, at the same time as new AP are gained. Using a Hero Card instantly refreshes the tavern, giving all new heroes and setting the countdown timer back to 60 minutes (but doesn't add AP). A refresh using a card seems to generate better heroes than a natural refresh.
Tip: It is worthwhile to build a tavern up to level 10 as soon as possible. Your initial heroes won't be very good (my level 43 combat hero from early on has lower combat stats [Att+Def+Spd] than a level 1 green hero from my tavern. This can make an important difference when battling other players. Don't forget that hero level is also very important for the troop limit, so don't discard your main hero.
Tip: It may be worthwhile to save up any Hero Cards gained until you have a level 10 tavern. If you have the points, it is certainly worthwhile to buy a Hero Box (containing 20 hero cards) and using them all immediately. Experience pills can also be saved for this event.
Arena
The infirmary is located in the Arena, where 75% of defeated troops will go to. You can pay gold to heal your units and use them again. Upgrading the Arena will speed up the time it takes to heal units by 1%.
The Arena also has a PvP arena where you can earn honor. However, you can lose almost half the troops that die in the Arena. Also, honor isn't fully implemented yet.
You can also fire and allocate troops through the Arena, though it's faster to use the units button on the hero tab to allocate them.
Warehouse
The Warehouse increases the amount of resources you can hold. You typically need it to keep it upgraded to collect enough resources for upgrading your Townhall.
Magic Tower
The Magic Tower has spells that can be researched. Some of them are required for unlocking new units. The Magic Tower has one tab for increasing various aspects of your kingdom and another primarily for battle spells. No extra bonuses for upgrading, it only unlocks another level of your spells for you to research.
Embassy
The Embassy allows you to access guild features and the guild leader needs to upgrade it to raise the member cap of the guild. The Embassy also raises the amount of defensive heroes your allies can send to you, to protect you from an attack.
The Market
Only some of the Market's features are implemented. The Market lets you swap your own resources around and sell them for gold. However, you cannot buy resources with gold. It can be a good idea to not buy any gold mines and just sell resources for gold when you need to, as it only goes one way. Currently, the Market shouldn't need to be upgraded for anything.
Bazaar
The Bazaar sells items and spell scrolls. The items are not as good as the ones you can collect from the worldmap, but the spell scrolls are special and can be useful at higher levels. Each level of the Bazaar unlocks higher level items that can be bought.