The IOTA GUI makes it possible to choose between Full Wallet, and Light Wallet. The Full Wallet automatically runs an IRI (IOTA Reference Implementation) instance in the background, which in turn means that you need neighbors in order to participate in the network and be synchronized. If you run the full wallet, you have no "trust requirements", as you are completely and independently participating in the network.
The Light Wallet makes it possible to connect to a remote node (whether it is your own node, or someone else's), and it's uses to get the latest state of the network, most importantly branch/trunk transactions which are required for making a transaction. When running a Light Wallet, it should be noted that your seed never leaves your wallet, and all the sensitive work (such as signing) is done client-side.