Triple DES (3DES) is an adaption of DES to use two keys. It follows the following algorithm, where the encryption function is that explained in the bullet points on the page about DES.
This triple-encryption can be proven to increase the security, and it is equivalent to doubling the length of the DES key (56-bits) to 112-bits. Cryptographers like to make sure their codes are secure by imagining the worst case scenario... As Janne Frösen puts it:
At a rate of one million keys per second, an exhaustive search of 2112 keys would require about 1.65 × 1020 years to complete. Since the universe is estimated to be only about 1010 years old, that is probably long enough for most purposes.
3DES is therefore pretty secure, and is used throughout the world, by governments, the military, banks and on the Internet. Since 3DES is based on DES, which has been around since the 1970's, it is thought that since no-one has found any shortcuts to breaking DES, there probably aren't ant to be found. However, no-one can prove that DES can't be broken by a quicker method, which means the possibility must remain that it can be. It is even possible that someone working secretly for the government has already broken DES, in a similar way that Enigma was broken by the British in the Second World War without anyone knowing...
There are two small problems with 3DES. Firstly it is about three times slower than DES, but since this is still pretty fast, especially compared to RSA, it is not too much of a worry. The second problem comes back to the keys. 3DES still has one key which is used for both encryption and decryption, and so this must be kept secret. We still can't use 3DES on its own for the Internet since there's no possible way to distribute keys securely for all the people wanting to use the Internet.