Typically this means having a separate card that plugs into a PCI slot in a computer that contains one or more coprocessors able to handle much of the SSL processing.
TLS accelerators may use off-the-shelf CPUs, but most use custom ASIC and RISC chips to do most of the difficult computational work.
During the TLS handshake the server and the client establish session keys (symmetric keys, used for the duration of a given session), but the encryption and signature of the TLS handshake messages itself is done using asymmetric keys, which requires more computational power than the symmetric cryptography used for the encryption/decryption of the session data.
Sometimes data centers employ dedicated servers for TLS acceleration in a reverse proxy configuration.
Modern x86 CPUs support Advanced Encryption Standard (AES) encoding and decoding in hardware, using the AES instruction set proposed by Intel in March 2008.