What is the difference between SHA-3(Keccak) and previous generation SHA algorithms? SHA-1 and SHA-2 share the same structure and mathematical operation as their predecessors - SHA-0 and MD5.  Both SHA-0 and MD5 have been broken. This is one of the main reasons why SHA-1 is considered insecure and is being phased out for SHA-2.

NIST announced in 2007 a competition to produce the next generation NIST hash function: SHA-3.

What is the major difference in structure and mathematical operation between the previous SHA-1 and SHA-2 functions and SHA-3?
They are all hash functions. Apart from that, they are structurally quite different. The SHA family (SHA-0, SHA-1, and the SHA-2 functions such as SHA-256 and SHA-512) use the Merkle-Damgård construction, around an internal permutation which happens to be an extended Feistel network. Low-level primitives include boolean bitwise operations, and addition over 32-bit or 64-bit integers.

In SHA3, the structure is that of a sponge function, which is quite different. Moreover, the inner permutation is not a Feistel network; low-level primitives are boolean bitwise operations over 64-bit words, but not additions. This avoidance of operations that involve carry propagation helps a lot with performance on hardware implementations (FPGA, ASIC).