To mine Obsidian, you need a diamond pickaxe. Nothing else will do. Also, Obsidian cannot be destroyed by TNT, so if someone tells you that Obsidian is overrated, it is probably a creeper. It can, however, be destroyed by the Wither.
How to make Obsidian
To make Obsidian, you need a lava source block. Put it into a cavity (meaning: a place 1 block in size surrounded by other blocks, e.g. dirt/cobblestone/glass/somethingwhichdoesn'tburn/Obsidian), then pour water over it. If you pour water over a lava flow instead, you get cobblestone.
You could dig deeply down to where the lava is and divert an underground river there, often you'll even find that this already happened naturally. Or pour a bucket, a nice way to close the lava so that you can mine the ore right above it. But mining Obsidian down there is tricky because often, there's more lava directly below the blocks, and you'll often lose the precious blocks to the lava when you mine them. Not to speak of the health hazard that'll be upon you some day, right, Notch?
How to make an Obsidian factory
To make an Obsidian factory, you need: A bucket (or two) and some sort of lava source (like a lava pool)
As said above, you need to put a lava source block in a 1x1 cavity and pour water over it. So, the most efficient way to go is: Make a checkerboard of cavities on the ground (out of whatever non-burning material), fill every hole with a lava source block, later pour a bucket of water over the whole thing.
Sometimes, picking the water source back up doesn't work (even though the bucket is full again), so just make a dirt block there, the water will subside then. Also, sometimes, the lava block will disappear thanks to lag. Just try again, maybe wait a short while before you do so.
How to make Obsidian in place
Instead of mining every block from whatever source, why not create it right at the spot where you want to place the blocks? Depending on where you do this / what you want to build / how you go about it, this can be much quicker than mining Obsidian from a factory (that you have to fill first) and then placing the blocks somewhere else.
This is of course a bit dangerous since you're handling lava and water. Lava... well, you know. And water can destroy redstone circuits, and if you have a nice lava moat nearby, you'll experience ultra-grief. Have fun mining all that Obsidian and refilling the moat with lava source blocks. I wonder if it will happen to my house (which has a lava moat) after some genius tried to make Obsidian on location nearby (because of this HowTo, to top it off) - accidentally tossing some water in the process. Dude, I'll come after you!
How you go about this of course depends on the location and the shape that you want to build / build on. I started doing this in the basement of my house, which is right at the bedrock, so it's the safest location ever to try this. (The permanent lava and water sources I made in my all-Obsidian basement look great.) Dirt will be an important tool. And you should use two buckets, one lava, one water.
You can place a lava source block and then quickly place a water source block right next to it, then you can pick the water source back up even before you get to see it. Tadaa, one block of Obsidian built! Sometimes, Internet transmission lag will interfere with this and you can end up with several unwanted outcomes, none of which should be fatal if you properly secured everything with dirt blocks before you started. By the way, it's slightly safer/easier to not perform this procedure in an "open field" (with all but one direction blocked), but instead make it in a "moat" situation, so that at least the lava block that you're placing is surrounded on three sides. I've had better experiences with this than with the all-open method.
And if you want to make a whole Obsidian ceiling above you, which I am currently doing, you have to first make a temporary dirt floor 1 block below, then kinda go about it like making an Obsidian factory: Make a checkerboard pattern of dirt, fill every hole with lava, pour water. Dig away the dirt checkerboard, fill with lava, pour water.
How to make a lava moat
If you tried to make a nice lava moat around your house, you have probably noticed that it is not as easy as you would think to place all those lava source blocks: They vanish. All you get is one lava source, the rest is flowing lava.
The trick is to not let the lava flow. Place a dirt block next to the first lava source block that you want to place. You can also put the dirt block into already flowing lava next to the source block. Later, dig away the dirt block (using the full lava bucket as a tool, so you don't have to switch fast), and when it's gone, immediately place the next lava source block. This way, you get your lava moat filled nicely with lava source blocks.
The moat, by the way, should best be made of - guess what - Obsidian, because then it's impossible to accidentally destroy or explode-away the container of your lava. Also, it looks great to have the near black Obsidian and the bright glowing animated lava combined.
Remember: You also need to contain the lava heat, so you shouldn't just make the absolute minimum moat casing without the corner blocks in the deep that no one ever gets to see, but invest more work and make those hidden corner blocks. To prevent unpleasant surprises.