Double-DES is two successive DES instances, while Triple-DES is three successive DES instances.

We use 3DES and not 2DES because 2DES does not yield the security increase that you would believe. Namely, 2DES uses 112 key bits (two 56-bit DES keys) but offers a security level of about 2⁵⁷, not 2¹¹², because of a "meet-in-the middle attack" which is well explained there (not to be confused with "man-in-the-middle", a completely different concept). Similarly, 3DES uses 168 key bits, but offers "only" 2¹¹² security (which is quite sufficient in practice). This also explains why 3DES is sometimes used with a 112-bit key (the third DES key is a copy of the first): going to 168 bits does not actually make things more secure.

This can be summarized as: we use n-DES because a simple DES is too weak (a 56-bit key can be brute-forced by a determined attacker), but in order to really improve security, we must go to n ≥ 3. Of course, every additional DES implies some computational overhead (simple DES is already quite slow in software, 3DES thrice as much).

Use AES.

In more details:

• DES is the old "data encryption standard" from the seventies. Its key size is too short for proper security (56 effective bits; this can be brute-forced, as has been demonstrated more than ten years ago). Also, DES uses 64-bit blocks, which raises some potential issues when encrypting several gigabytes of data with the same key (a gigabyte is not that big nowadays).
• 3DES is a trick to reuse DES implementations, by cascading three instances of DES (with distinct keys). 3DES is believed to be secure up to at least "2¹¹²" security (which is quite a lot, and quite far in the realm of "not breakable with today's technology"). But it is slow, especially in software (DES was designed for efficient hardware implementation, but it sucks in software; and 3DES sucks three times as much).
• Blowfish is a block cipher proposed by Bruce Schneier, and deployed in some softwares. Blowfish can use huge keys and is believed secure, except with regards to its block size, which is 64 bits, just like DES and 3DES. Blowfish is efficient in software, at least on some software platforms (it uses key-dependent lookup tables, hence performance depends on how the platform handles memory and caches).
• AES is the successor of DES as standard symmetric encryption algorithm for US federal organizations (and as standard for pretty much everybody else, too). AES accepts keys of 128, 192 or 256 bits (128 bits is already very unbreakable), uses 128-bit blocks (so no issue there), and is efficient in both software and hardware. It was selected through an open competition involving hundreds of cryptographers during several years. Basically, you cannot have better than that.

So, when in doubt, use AES.

Note that a block cipher is a box which encrypts "blocks" (128-bit chunks of data with AES). When encrypting a "message" which may be longer than 128 bits, the message must be split into blocks, and the actual way you do the split is called the mode of operation or "chaining". The naive mode (simple split) is called ECB and has issues. Using a block cipher properly is not easy, and it is more important than selecting between, e.g., AES or 3DES.