As in the English 'I love you' and 'I tell you', ti may mean 'to you' or 'you' depending on the verb.
'to you' when the verb is intransitive (ie needs a preposition). Ti parlo = parlo a te.
'you' when the verb is transitive (ie doesn't need a preposition when answering the question 'who/what'). Ex. Ti amo = amo te
I'm not sure about the second part of your comment. What is "this"? Ti is the pronoun for tu which is used when you are talking to/with one person you are in familiar terms with.
If that was a group of people, you'd need to use voi.
If it was a person you are not familiar with or a person you have respect for, then you'd use Lei'.
Well, the subject is 'I'.
This is the conjugation of volere:
The pronouns (io, tu...) are in brackets because they are rarely used in real life.
Moreover you have ti which means 'you' as object (like 'I'm thinking of you'), never subject.
'you want me' would be mi vuoi.