The idea - Heavy mine usage for defensive and offensive purposes, similar to broodwar vulture vs dragoon aggressive mining style. Tanks to protect the mines from stalkers/other ground based unit and also to provide long ranged constant damage vs buildings. Marines to support the tanks, provide anti-air/observer services and just decent fodder + dps unit.
The build -
10 depot 12 rax 13 gas, 15 OC, 15 marine, 16 gas (right after the marine, with the scv building the rax), then depot again
Make 2 marines and throw down fact + reactor on rax. Factory again @ 100 gas. When 1st fact finishes, switch onto the reactor of the rax and proceed to make mines. Land the rax and make another reactor - make marines from these. When second fact finishes, make a tech lab and proceed to make tanks.
Chances are you will not be able to afford all of these right at the start, so skip the marines when the reactor on the rax finishes and grab the tank first. Mines + tanks should always be constant. DO NOT GET SUPPLY BLOCKED.
If opponent is doing some kind of all in/stargate play - the amount of mines you have should easily shut it down. Tanks will help deal against any 4 gate/4gate blink rushes. Marines are there to snipe the DTs.
Pushing out - When your second tank comes out, throw down an ebay and grab a 3-5scvs (vs T) or 5-8 (vs P) for repair + turret placing purposes. You can push out much earlier if you know what you are doing, but against decent gateway units you need at least two tanks (tanks 4 shot workers and sentries). You must try not to lose any tanks and mines (marines too but thats really damn hard), as this is a pure dps build and losing any units decreases its power heavily.
Positioning of the tanks is also extremely important. The Nexus cannon is the same range as siege tanks in siege mode, so if you think its possible to push up directly into the main and avoid the nat cannon all together - do so. If you cannot, you will end up containing the guy for a long ass time until the cannon runs out - depending if your opponent is any good, this build will either flat out kill him (unless you messed up really badly), or you will have to expand and transition as you see fit.
The killing blow usually happens if you manage to get two tanks and 6+ mines inside his main, with your scvs building turrets inside his main. Your tanks will be in range of a lot of his tech buildings and preventing him from warping in a lot of units (sniping pylons).
However, as I said, its not that great vs protosses as they learn how to deal with widowmines more and more.
I have tried using thors instead of tanks, but thors ended up being way too bulky and hard to micro. Repairing scvs would block their pathing, preventing the thors from moving at all (even with only three scvs)
Vs T, I have NOT have any problems with this - the reasoning is that the build is extremely safe vs any gas openers (especially the hellbat drop - you will have 6 mines and 2.7 siege tanks with a handful or marines by the time the drop comes. Pair up the mines and burrow down possible drop edges in your main, if they still get in, tanks will make short work of the remaining drop.
Against macro builds - a smart terran who scouts out the buildings (kr masters) will have a bunker up front and marines in the main ready to intercept possible drops. A not so smart terran might have a bunker up front and a group of marines near the bunker. In either situation, all you have to do is forcifully burrow the 8 mines hugging the bunker, snipe down the bunker with the marine + tanks, siege up in a nice spot immediately once the bunker goes down. If his reaction is super super super fucking fast he will back off immediately giving you insane coverage over his nat with the siege tanks, immediately giving you access to his entire nat mineral line. The build hits before stim, and since in TvT stim s always researched first - no combat shield OR stim by the time the build hits. This means widowmines will one shot a chunk of marines if the opponent is not careful, and the two tanks will also one shot a nice chunk of marines.
This build does NOT work vs Z do NOT attempt to use this aggressively with it.