Your Bitcoin wallet is what allows you to transact with other users. It gives you ownership of a Bitcoin balance so that you can send and receive bitcoins. Just like email, all wallets can interoperate with each other.
Software wallets are installed on your computer. They give you complete control over your wallet. You are responsible for protecting your money and doing backups.
Mobile wallets allow you to bring Bitcoin with you in your pocket. You can exchange coins easily and pay in physical stores by scanning a QR code or using NFC "tap to pay".
Web wallets allow you to use Bitcoin anywhere with less effort to protect your wallet. However, you must choose your web wallet with care as they host your bitcoins.
