What does John 14:14 mean?

This is a concise summary of the prior verse (John 14:13). Despite being widely misinterpreted, Jesus does not give a blanket promise to do whatever we want, whenever we want it. The context is reassurance to the disciples (John 14:1–11), and encouragement to hold fast in the face of hardship. It's also tied to Christ's promise that those who follow Him have opportunity to do "greater" works (John 14:12).

A key condition here is asking "in [His] name." To act "in the name of" someone means to invoke their authority and their will, rather than your own. An ambassador who offers an agreement "in the name of the king" is enacting the decision of that king. A police officer who says, "stop, in the name of the law!" is—or should be—acting out the will of the government, not their own opinions.

To pray "in Jesus' name" is not evoking a magical spell, as if the words themselves have power. It is possible to use that phrase and not, in fact, be praying "in the name" of Christ. This is just the same as a police officer who demands a bribe can't do so "in the name of the law." Those who refuse to offer a bribe can't be legitimately arrested. Anything we ask for in prayer must be according to the character of God, and in the will of God, if it is to be granted.

This condition connects to the statement Jesus makes in the next verse: that those who claim to follow God prove it by their actions (John 14:15).