A wallet can be as simple as just a private key by itself even as software that has access to that can derive the public address from it. So yes, that data alone can be considered a bitcoin wallet.
But you asked if that is all you need to send and receive so for that you will need software that uses the private key to spend the funds. Sometimes the term "wallet" is used to refer not to just the data stored but transaction data as well, along with the software that can create transactions and do the accounting for them as well so that you can know your balance.
Proper vocabulary would describe wallet software as being a bitcoin client. The bitcoin client from Bitcoin.org (also referred to as the "Satoshi" client) does this, and has a local wallet as well.
Another type of wallet is an e-wallet where the wallet data is stored by the service's operator. An e-wallet can also be referred to as a "hosted wallet". InstaWallet is one of these. The just-launched Paytunia is another. An account at an exchange which hold bitcoins for you is another type of e-wallet service. Using a hosted wallet incurs risks involving trust extended to the wallet's operator.
But the simplest form of a wallet comes from http://BitAddress.org
It gives exactly what you describe -- a private key and a public address. This happens to provide one of the most secure wallets as well. The html source can be copied to a thumb drive and then used to create paper wallets from an "air gapped" system (such as a computer running a secure operating system perhaps using a LiveCD linux distribution) that has no connection to the network and is used for printing out paper wallets.