Taproot Is Coming: What It Is, and How It Will Benefit Bitcoin

P2SH

All bitcoins are essentially “locked up” in scripts: a couple lines of code embedded in a transaction included in the blockchain, that define how the coins can be spent in the next transaction. Spending conditions usually involve providing a signature to prove ownership of the coins. But other, well-known conditions for example include timelocks (coins can only be spent after a specific block height or date) or multisig (coins can only be spent if some number of private keys out of a set of private keys provide signatures).
MAST

MAST (Merkelized Abstract Syntax Tree) is a proposed solution that uses Merkle trees (a decades-old, compact data structure invented by cryptographer Ralph Merkle) to work around these two downsides. In short, all the different conditions under which the funds can be spent are individually hashed (as opposed to combined into a single hash) and included in a Merkle tree, which ultimately produces a single hash: the Merkle root. This Merkle root “locks up” the coins.
Schnorr

The Schnorr signature scheme has long been on the wishlist of many Bitcoin developers and is currently in development to be deployed as a soft fork protocol upgrade. Many cryptographers consider the Schnorr signature scheme to be the best in the field, as its mathematical properties offer a strong level of correctness, it doesn’t suffer from malleability and is relatively fast to verify.
Taproot

Taproot is based on an interesting realization: no matter how complex, almost any MAST-construction could (or should) include a condition that allows all participants to agree on the outcome and simply sign off on a settlement transaction together. In the earlier example, if Bob knows Alice can, by herself, claim all the funds next week, he might as well cooperate with her now to sign off together. (In many typical smart contract setups he would even be penalized if he doesn’t. The complexity really just serves to keep everyone honest.)

full post or source- https://bitcoinmagazine.com/articles/taproot-coming-what-it-and-how-it-will-benefit-bitcoin