I am kinda of confused about what are some major differences between AES, Twofish, and other types of common encryption algorithms.
Rijndael (aka AES) and Twofish were both candidates and finalists for the Advanced Encryption Standard contest, a three year selection process which yielded the selection of Rijndael as the standard.
Contest submissions were required to be block ciphers of block size 128 bits and support key sizes of 128,192 and 256 bits. Submissions were put through rounds of cryptanalysis and assessment by prominent members of the cryptographic community. NIST, the competition organiser, made the final decision that Rijndael would become the AES.
At a high level both AES and Twofish are 128 bit block ciphers supporting 128,192 and 256 bit key sizes. Both ciphers are (despite a small number of theoretical attacks) secure in the computational/pragmatic sense that nobody has yet found a way to break them. Both ciphers are based on heuristic constructions, meaning that we don't have a proof of security showing a reduction to some known hard problem, instead their security is based on the fact that nobody has broken them yet. Both ciphers satisfy the pseudorandom permutation (PRP) function model in that their output cannot be distinguished from a random permutation in the block size.
High-level differences between the ciphers are that AES and Twofish are based on a substitution-permutation network (SPN) and a Feistel-network respectiveley. These networks are applied in rounds where AES has 10,12 or 14 rounds depending on the key-size, Twofish always applies 16 rounds.
Aes is based on elliptical curve and abstract mathematics such as groups and fields. Twofish isn't. Aes is commonly used by most https servers and is recognized by ssl or tls, twofish isn't.