I think that in answering this question, it's important to understand that the differences in bottom brackets themselves is partly due to the cranks that match said bottom bracket 'doing more of the work', so the crank and BB need to be seen as a system for the whole picture.
One-piece cranks ("Ashtabula")
In this system, the crankarms are a single piece of metal bent into a "Z" shape, with threading on parts of the spindle section of the crank (the part that goes through the frame). Bearing cups are pressed into the frame, then the crankarms are inserted through in one piece. bearings go between the frame cups and cones that thread onto the crank spindle. The Park Tool One-Piece Bottom Bracket service page has lots of nice pictures that might help figure out how things go together.
Three-piece cranks
These cranksets consist of a left and a right crankarm, which both attach onto a spindle which is part of the bottom bracket. Usually these BBs will thread into the frame, with the majority of the assembly being one piece that threads into the drive- (right-hand) side of the frame, and then a (usually plastic in the case of Shimano bottom brackets) supporting piece will thread into the non-drive-side to take some of the load off the drive-side threads.
Examples of spindle systems for three-piece cranks are square-taper (both ISO and JIS standards), Shimano Octalink splined and ISIS splined. These BB standards all thread into the frame. The crankarms usually attach to the spindle via a bolt that threads into the end of the spindle.
Two-piece cranks
In recent years, most higher-end cranksets have switched to a construction where the spindle and drive-side crank/spider assembly are integrated into a single unit. This means that the bottom bracket no longer needs to have a spindle, and is now essentially a pair of bearings that support the crank spindle. The variation in where these bearings are placed, their size, and how they are mounted in the frame causes the wide range of modern bottom brackets.
Narrow-diameter spindles (≈ 24 mm)
Shimano's Hollowtech II and FSA's MegaExo both use a spindle diameter of 24 mm, and are most commonly used with external bearing BB's that thread into the frame, such as the SM-BB70 you mention.
SRAM's GXP system uses a 24 mm spindle on the drive size which narrows to 22 mm on the non-drive side, which they say avoids having to adjust the axial loading ('preload') on the bearings during installation.
Large diameter spindles (30 mm)
In the search for lighter and stiffer cranks, the diameter of the spindle was increased to 30 mm from the previous 24 mm. However, the cartridge bearings for a 30 mm spindle have an outer diameter that is too large to be attached with threaded external bearing cups, so most new systems moved to 'press-fit' BBs. There are a number of standards, the first of which was Cannondale's BB30, where the cartridge bearings are pressed directly into a machined bottom bracket shell in the frame.
However, because the manufacturing tolerances required to achieve a good fit for the bearings in a BB30 system are very tight (and therefore expensive to manufacture), other manufacturers switched to PF30, which uses the same cranksets as BB30 but rather than pressing the bearings directly into the frame, plastic bearing cups are pressed in first, and then the bearings are pressed into the cups.
Mixtures
Very recently, FSA have introduced their BB386EVO cranks, which have the same diameter spindle as BB30/PF30 cranks, but the spindle is wider. The wider spindle means that FSA can offer a BSA threaded frame to 30 mm crank bottom bracket, but that the crank can still fit to BB30/PF30 just by adding spacers to the spindle to make up for the narrower BB width of those standards.
There are also standards, which I do not know as much about, which concern press-fit bearings sized to accept 24 mm cranks, including "Shimano press-fit" (colloquially BB86 or BB92), and Trek's BB90, which uses slip-fit bearings like an integrated headset, as opposed to bearings that must be pressed in with force.
Campagnolo
Campagnolo produce three types of cranksets at the moment. Power Torque is their name for a two-piece crank similar to Shimano Hollowtech, however the bearings are not part of the bottom bracket, instead they are pre-installed on the crank spindle before it is inserted into the frame.
They also produce Over-Torque, which is a 30 mm spindle design compatible with BSA threaded frames and most press-fit systems, and Ultra Torque which splits the spindle in two at its midpoint.
(Apologies for the lack of detail in the Campy section.)
