Romeo is quite young and privileged, and in him, Shakespeare has painted a portrait of adolescent angst and desire. Romeo is just learning to come to grips with his sexuality as well as with the exuberant and painful emotions that are emerging as he moves from youth to adulthood. He is grappling with feelings of romantic love and the power and possibility of steering his own course. Everything for him is experienced at the peak of intensity, and like many adolescents, tomorrow feels like an eternity away: he must have everything right now or he feels he can't go on. Although this is a tragedy, Shakespeare has some fun with how thoroughly and dramatically Romeo throws himself into the moment. As the play opens, Romeo, who is from a wealthy enough family that he doesn't need to work, moons and mopes over Rosaline, believing he can never ever love another woman—until he meets Juliet and immediately falls headlong in love with her.
The friar counsels Romeo in Act II to exercise some restraint and moderation, saying to him:
These violent delights have violent ends,
And in their triumph die; like fire and powder,
Which, as they kiss, consume: the sweetest honey
Is loathsome in his own deliciousness,
And in the taste confounds the appetite:
Therefore love moderately ...
Romeo may have developed that kind of longer view had he had the chance to grow older and more mature. Experience would teach him that he doesn't have to act on every emotion and impulse immediately. But tragically, Romeo never has the opportunity to fully grow up.
I suppose one of the key things that is traditionally seen to mae Romeo immature is the way he suddenly changes his romantic desire from Rosaline (an off-stage character, never seen, whom he loves as the play opens) to Juliet - based, of course, on a single glimpse.
Moreover, Romeo is hugely - almost comically - impetuous: look at the way he insists that Friar Laurence marries him and Juliet the very same day; and look, more interestingly, at the way he murders Tybalt immediately after hearing of Mercutio's death. Friar Laurence repeatedly says that his reaction is immature and "womanish" - tears and tearing out his hair: but the key point is that Romeo ruins his own future in Verona (and Friar Laurence's plan that the wedding between R+J will heal the Capulet and Montague feud). And for no real reason - no real gain.
Perhaps the most immature thing about Romeo then is his inability to think before he acts: and indeed, his suicide (entirely unnecessary as it turns out: Juliet is alive) seals the tragedy of the play.