Running a full node gives you direct access to the Ethereum blockchain. If you do not have a full node, you will need to connect to another node to get the necessary information and verify your transactions.
To run a full node Ethereum wallet, you need to download the entire Ethereum blockchain. Basically, this turns your computer into a network-supporting node that negotiates with other nodes and validates transactions. Because blockchain is gigabytes in size and constantly growing, this is only possible with a computer wallet.