Тебя-то

Тебя-то мне как раз и нужно
What is this?: Тебя-то

Particle "-то" here has no meaning itself, it just stresses the word "тебя".

-то is used to mean 'It is YOU who I need'.

-то is something like definite article the

The entire "Тебя-то мне как раз и нужно" is used in such circumstances : you have planned to contact someone in near future but suddenly come across the person incidentally.

Vchera vecherom ya skazal devushke o ee koshke:
"Tvoya-to ochen upryama ... ne xochet yuiti otsiuda" (koshka lezhala na divane)

Correct context and use?

"-то" puts a lot of logical stress on "Твоя". That should be done for a reason, because normally stress would be on "упряма". Probably, if you discuss some other cats and then suddenly change topic to her cat you can use "-то" here.