Just like in other major cities, getting to know the public bus system can be quite a hard task. The Tokyo Metro Government runs most of the buses within inner Tokyo (Going by the name "Toei Bus") while a number of private operators run services in the suburbs.
For nearly all places, you'll have no problems doing a trip with the use of the JR or Metro trains only, but for those who really want to travel like a local, here are some tips:
How to use them
Board from the front. Put your fare (a flat fare of 210 yen) in the box next to the driver. If you have a 1,000 yen bill, there is a slot where you can insert your bill and it will automatically give back your change. There is no ticket or receipt, simply proceed further and find a seat.
(Note that Toei is the only operator that has the auto-change function. Most other operators have just a money changer that will break down your bill to smaller coins. Also any bills over 1,000 yen can't be used so be careful!!!)
If you have a PASMO or SUICA card, you can use them also; just tap on the sensor located next to the driver when you board. You don't need to touch it a second time when getting off.
When you want to get off, push a button near you and exit from the door in the middle (rear) of the bus.
*When you are traveling outside central Tokyo, it may not be this easy - it's usually not a flat fare and the rate will be determined based on the distance you travel. In this case, you enter from the rear and take a small ticket which has a number on it, and then pay as you leave from the front. There will be a screen at the front of the bus that tells you how much you have to pay. If you have a PASMO/SUICA, you tap once when getting on and again when getting off and the fare will be deducted automatically.
Which routes to use
Now here's the tricky part - knowing which routes to use. What's confusing for non-Japanese readers is that the routes have a prefix in Chinese character (kanji) followed by a two-degit number. For instance, at Shimbashi station there is a 都01 going to Shibuya Station and a 市01 headed for Tsukiji Market! There is small English signage of the destination at the front of the bus, so you'll have to examine that carefully.
Of all the routes out there, 都01 running between the Shibuya Station east bus terminal and Shimbashi Station stops by the Roppongi Hills which is convenient since you would have to change at least once if using trains. The RH01 runs a direct route between Shibuya and Roppongi without making stops, so it is the fastest way to travel between the two locations.
The 都08 originates at Kinshicho Station's north terminal (connection to JR and Metro Hanzomon Line)and runs along the boulevard south of the Tokyo Skytree, with stops at the Asakusa Station before terminating in Nippori Station which is a good base for exploring the neighborhood of Yanaka.
The 海01 routes starts at Monzen Nakacho Station of the Metro Tozai Line. It makes a stop at Toyosu Station (Metro Yurakucho Line) before ending up in the Odaiba District of the waterfront. It's a cheaper alternative to the expensive Yurikamome Line, though the views will be far less impressive and would take double the amount of time.
All of the above routes have a frequency of one service about every 7 or 8 minutes; other routes might have less with only 2-3 services an hour. The buses stop running far earlier than the trains, with the last one around 10pm.
The Tokyo One Day Ticket (1,590 yen) or the Toei One-Day Economy Pass (700 yen) will work on the Toei Buses, but the Tokyo Metro Pass will not.
There are some additional info available on the Toei English site. http://www.kotsu.metro.tokyo.jp/bus/routes/eng/