I'm really surprised that nobody has commented on this with ayaat and ahadith. This is not a complicated topic that you study for years to realize the right answer. This is clear-cut.
Lets go there now. I will briefly mention some of the things that are mentioned when this topic came p in various classes/lectures/conferences/whatever. The summary is,
the only girlfriend you're allowed is your wife! 
(And of course, everything applies symmetrically to both genders).
- Allah says in the Qur'an: laa taqraba az-zina [
Surah Al-Israa, verse 32] which means "do not come CLOSE to zina (a man/woman relationship outside of marriage)." Let alone actually having said relationship.
- There is a statement (I thought it was a hadith, but I can't find the source) that "whatever leads to haram is haram."
- There's a hadith that states that it is better to be stabbed with an iron rod than to touch the hand of a non-Mahram of the opposite gender (source uknown). There are
many related hadith in Bukarhi and Muslim that indicate that rasulullah (salallahu alayhi wa sallam) never touched the hand of a non-mahram woman -- not even in the most important event of all, MORE important than shaking the hand of your boss, which is when he took the pledge of Islam from various groups.
- There's a hadith that "when two (a man and woman) are alone, shaytan is the third." [
Tirmidhi]
- One of the five high-level goals of Islam is protection of lineage. If people can have relationships with anyone, lineage becomes confused. You can see this in western countries where you have soap-opera-like tales of "who's the dad."
This is only what I can remember right this second; I'm sure there's more. It's clear that Islam doesn't allow this. Look around (if you live in a western country) and you can see all the problems that come up when people don't follow these rulings.
If you're looking for a girlfriend, look no further than marriage

it's more stable, but more responsibility; and ultimately, it's THE halal option available to us.
Wallahu ta'ala a'lam.