This pattern is used to talk about your own intentions or plans. It's a natural way to say 'I think I'll do...' or 'I'm planning to do...'. The verb is conjugated to its volitional form (the same form used for 'let's do' or 'shall we do'), then you add と思う (とおもう).

It's more definite than just saying つもりです, because it includes the nuance of 'thinking' about it. It's often used when you're making a decision on the spot or announcing a plan. For example, if someone asks you what you'll do this weekend, you might say '映画を見ようと思う' (I'm thinking of watching a movie).

This pattern is mainly used for the speaker's own intentions. For someone else's intentions, you would use と思っている (おもっている) or other expressions. It's common in both speaking and writing, but in casual conversation, the と is often dropped, like '行こうと思う'. In more formal speech, you might hear と思っています.