DES is defined to use a 64-bit key. Only 56 of these bits are really used, so the "effective key length" (for resistance against exhaustive search) is that of a 56-bit key. Yet, any implementation will expect a sequence of 64 bits (hence, 8 bytes, not 7).
Triple-DES (aka "3DES") is three DES instances in due sequence. The "middle" DES instance is used in decryption mode, precisely so that an engine which implements 3DES can also execute plain DES. If you name the three successive keys K1, K2 and K3, then, when K1 = K2, the two first DES instances cancel each other (the second DES being in decryption mode, it reverses what the first DES did), so this is equivalent to simple DES with key K3. Similarly, if K2 = K3, this is equivalent to simple DES with key K1.
3DES nominally uses a 192-bit key (three 64-bit DES keys), out of which 168 bits are really used. Yet, there is an "academic" attack against 3DES with cost 2112, so it is often said that the overall security of 3DES is similar to that offered by a theoretically perfect block cipher with a 112-bit key. Hence, there is a widespread usage mode of 3DES in which we use a 128-bit key: 64 bits for K1 and 64 bits for K2, and then set K3 = K1. In plain words, encrypt the block with K1, then decrypt with K2, then encrypt again with K1. This seems sufficient to achieve the 112-bit level of security (of the 128 key bits, only 112 are really used), and the academic attack shows that you cannot go beyond that level anyway (112 bits are already quite far beyond what is technologically breakable right now, so no worry)(Edit: D.W. points out that there is a better attack, provided that you can harvest quite a few plaintext/ciphertext pairs; still, the attack cost is still too high to be used in practice). This is what your smartcard implements.
Note that there are protocols which use 3DES and really require the full 192-bit key, and cannot work with an implementation which only supports 128-bit keys; a prime example is SSL.