A wallet is the place where you can store your hard-earned Chia. The official Chia Blockchain comes with a built-in wallet, where XCH coins that are sent to any of your public addresses (under that private key). This wallet requires you to sync to the Chia Blockchain to see your total balance, but can be used even when partially synced.
Other wallet solutions are being built out. Nucle.io is the first in this space. Nucle is the world’s first open-source, fully decentralized Chia blockchain “Simplified Payment Verification”(SPV) type wallet. They sync their node, so you don't have to. This means that you can instantly have access to your Chia wallet, without the need to sync.
Another wallet that is currently being developped is the Arbor wallet, that launched as a discord bot, but is being built out as we speak. An introductory video can be found here: https://www.youtube.com/watch?v=RCLrVWfz2Z8