For asymmetric encryption such as embodied by RSA, the "key length" is conventionally taken to be the length of one of the public key elements. Such a key is highly structured (only a very few sequences of bits of that length are valid keys) and security relies on robustness of this structure with regards to some specific analysis algorithms. E.g., in RSA, the "key size" is the length of a big integer (called the "modulus") and RSA is secure as long as one cannot factor that big integer into its prime factors. Integer factorization becomes hard, then unfeasible with existing technology, when we consider large enough integers. 1024-bit integers for RSA are commonplace, and it is recommended to use larger ones (e.g. 2048 bits -- cryptographers just love powers of 2), to account for possible advances in technology in the next decades. Implementation of RSA requires "big integers", i.e. integers with length not limited to that of the CPU registers. See @mikeazo's answer for pointers.
For symmetric encryption, it makes no real sense to use a key larger than 256 bits (arguably, 256 bits are already extreme overkill). Symmetric encryption algorithms accept as keys all sequences of a given size. The key size must be such that it is not feasible to try all possible keys, an attack known as "exhaustive search". 256 bits are already way beyond the reasonable and also the unreasonable; see this answer on security.SE for a more detailed analysis.
A non-weak symmetric encryption algorithm is such that if you do not use exactly the right key, you get utter garbage, so that you cannot approximate the key: you really have to hit it. Another way to say it is that for a non-weak algorithm, exhaustive search is the best attack. Yet another way to say it is that if enlarging the key beyond 256 bits makes any difference to security, then you are using weak symmetric encryption algorithms. It is a rather risky bet to assume that weaknesses in an algorithm can be overcome with larger keys; a much safer attitude would be to just stop using weak algorithms.
As for implementations, some cryptographic algorithms are defined with arithmetic operations on relatively large operands. E.g. RC6 uses multiplications of 32-bit integers. This is done because big processors (such as the core CPU of a PC) are good at such kinds of operations. However, smaller processors are not: smartcard developers, and people who design custom circuits on ASIC or FPGA are not fond of RC6. So other algorithms are built only on elements which are easy to deal with on any platform. E.g. the AES uses only 8-bit operations. There is no relation between the block size (128 bits for AES), the key size (128, 192 or 256 bits for AES), and the type of elementary operation the block cipher is built on.
Anyway, if an algorithm runs at all anywhere, then it is implementable with only transistors (what else is there in a CPU ?) and a transistor handles only 1-bit values.
