Segregated Witness is a proposed change to the Bitcoin code and network.
It fixes a number of perceived issues in the Bitcoin network.
There are two major Bitcoin Improvement Proposals that it includes:
SegWit includes a fix changing how a label (transaction id) is chosen when bitcoin is sent.
Currently the label can be changed by a node (someone relaying the transaction), someone who approved sending the transaction, or a miner (someone confirming the transaction) before the transaction is confirmed on the network.
By changing the label on a transaction it becomes harder for Bitcoin wallets to track payments. Additionally this can lead to failed payments if people try to spend unconfirmed bitcoins. Unconfirmed bitcoins are bitcoins included in a payment that haven't been added to a block yet.
The fix moves the modifiable parts of the transaction to the transaction witness and segregating it from the rest of the transaction. The rest of the transaction can then be used to generate a label that the Bitcoin network can use to refer to the transaction with, and it can't be modified.
There is an issue on the Bitcoin network where maliciously designed transactions can require more time to be verified. This issue scales if more transactions are included per block.
These two fixes, although seemingly small, could help more transactions verify faster, fixing some of the current blocksize issues. But any change, big or small, to the Bitcoin network needs thorough testing before it can be implemented.