I don't have too much interest in playing Terran, but I'd like this information mostly to better understand what I'm up against.
There are literally dozens of Terran builds and possibly hundreds of Cheeses. As a result I'm going to try and lay out a couple of the broad strokes, but unlike Protoss or Zerg these are not going to be succinct build orders.
The three major builds in Terran play are:
Obviously there is a lot of variation and overlap between these builds but I'm going to try and lay out the broad strokes here.
Terran bio revolves around building a Marine/Marauder/Medivac force for a nice timing push. This is sometimes called the MMM build. Openings usually consist of 3 Barracks (sometimes called Rax), 2 with techlabs and 1 with a reactor. A Terran economy can readily support 3 Barracks . The timing push here usually comes at about 40 food with a heavy Marauder/Marine force with Stim. The build continues to evolve into the late game by adding on a Factory and Starport for Medivac support. Medivacs are key to Stim based play, as any force that stims is necessarily at reduced HP for the remainder of the game. Additionally, Medivac's provide for drops which increases the mobility of your army. You can also transition into Bio mech, or add Ghosts on for TvP.
Terran Mech grew out of early hellion harass. This usually involved building one Barracks before going to 2 Factory and pumping out hellions for early harass and then transitioning into a more stable Thor/Siege tank force. This is especially effective vs Zerg players going Muta/Ling as upgraded Hellions are powerful vs Zerglings and Thors do splash damage to Mutalisk. Additionally, heavy Siege tank forces are effective vs Hydra/Roach armies. Top Zerg players (such as Artosis and IdrA) have said that Terran Mech is so powerful it is "Broken" (a term to mean unbalanced) against Zerg. A key point to remember is that the initial Barracks can be used for scouting or to produce Marines (which help an early Mech army).
The 1:1:1 was first popularized by the player The Little One (TLO) and has become the standard in TvT play. The name derives from having one of each unit producing structure: 1 Barracks, 1 Factory, 1 Starport. The synergy that units from these buildings provide have cause this build to become increasingly popular outside of TvT play. The basic idea here is to build a Marine, Tank, Viking force, using the Vikings to give Line of Sight to the Siege tanks (which can fire further than they can see) and using Marines to fight off Air harass.
One of the important points of this strategy is to use Vikings to establish air dominance. In a losing battle, the Vikings can always land and wait for Marine support (as Vikings are an air to air unit). The power from this build comes in its flexibility. Usually 1 techlab and 2 reactors, these addons can quickly be switched, allowing the deployment of Banshees (once air superiority has been obtained) and Hellions, or swapping to Marauders to break a Siege tank grouping. Unless there are serious balance changes in the future a number of SC2 Commentators (Huk, Day9) have suggested that this will become the standard Terran play.
I'll briefly mention a couple of the standard Terran Cheese strategies.
6 Rax into Reaper harass is a popular TvP strategy, as early reapers built from a proxy Rax (a Barracks built close to the opponent) will arrive before the Protoss has Stalkers and thus are unbeatable with good micro.
Banshee rush, especially against Zerg opponents has also been popular as Zerg does not have a staple unit before Tier 2 that can attack air.
Thor drops (often a part of mech builds, or biomech) are very effective. While rushing thor drops can be considered a Cheese, use of them as part of a normal build is present at even the competitive level.
Standard Terran opening is:
How to read these: The number indicates the number of Supply you should have when you build your first structure. This number is usually SCVs but may include other units.