We see in Alien how the Nostromo is diverted to the planet to investigate the crashed ship, but is that the first encounter with an alien species in that universe, or has the human race made first contact with other extra-terrestrial life forms previously?
In Aliens the marines talk about "bug hunts" which implies there are some native animals which have been killed by marines on some colonised worlds. The events of Alien haven't reached the world at large in Aliens (only some people at the Weyland corporation would know about it.)
Hudson: Is this gonna be a standup fight, sir, or another bughunt?
Gorman: All we know is that there's still no contact with the colony, and that a xenomorph may be involved.
Frost: Excuse me sir, a-a what?
Gorman: A xenomorph.
Hicks: It's a bughunt.
If this is known about in Aliens, it stands to reason it would have been known about in Alien too.
In Prometheus, which I think is meant to be set in the same universe as the Alien franchise, humans seek out and make contact with a race referred to as The Engineers (who may or may not, depending on how much sense you can extract from the screenplay, have bio-engineered the Alien from Alien).
In Alien vs. Predator (which, as @AnkitSharma pointed out, is not considered canon by Ridley Scott) the Weyland Corporation sends a team to the Antarctic to investigate Predators hunting Aliens in a buried pyramid. (I can confirm that absolutely nothing interesting happens in Aliens vs. Predator: Requiem.)
So, if either of those films are in the same universe, yes, but in both cases the contact did not seem to be made generally known to the world at large (and was at least somewhat related to The Alien).
However, even ignoring the quotes from Aliens mentioned in other answers, Alien pointedly does not feature any scenes where the Nostromo crew say "Holy f-, this planet has alien life! Humanity is not alone in the universe after all! What an incredible and paradigm-shifting scientific discovery this is!"
Maybe they just had other things on their mind (hazard pay, flirting with Lambert, trying not to get eaten), but it does imply that maybe, given the deep space mining they're doing, alien life of some sort has indeed been encountered even before the Nostromo did (let alone in the 60 years between then and Aliens), and the crew are aware of that.
The full Alien/Predator universe only has several sentient species:
Most of which are basically Predators (Yajuta), Aliens, Engineers and Humans (there's also River Ghost and Snorgesh but I don't think they were "contacts" as far as such things count).
As such, the history seems to be (based on the whole mess of timeline between AvP*, Prometheus, Predators, etc...):
- Predators: Came to Earth both prehistorically (AvP) and in 20th century (Predator*). Contact was classified.
- River Ghost - encountered in Predator preserve by huntees, beginning 20th century (Predators film)
- Engineers (Prometheus): Visited earth and interacted with Earthlings (including, supposedly, Jesus being an Engineer). Visited in 21st century by Prometheus.
- Aliens - encountered by Prometheus (presumably with no widespread knowledge given "Alien" events) and later by Nostromo
There's no mention of any prior extraterrestrial contact in pure Alien(s) franchise.
Aliens has this crude dialogue on this species:
Frost: “Hey, I sure wouldn't mind getting some more of that Arcturian poontang, remember that time?"
Spunkmeyer: "Yeah, Frost, but the one that you had was male."
Frost: "It doesn't matter when it's Arcturian, baby.”
Wikia has this:
The Arcturians are a humanoid, mammal-like extraterrestrial species native to the planet Arcturus. They were the first sentient alien species encountered by mankind and humanity has since established interspecies trade with them.
Arcturians are roughly humanoid, bipedal in form and comparable in size to humans. However, they are comparatively lacking in mental capacity; their intelligence has been likened to that of lower apes. Biologically, they exhibit an unusual combination of insect and mammalian traits — physiologically, they mammal-like, yet they live in subterranean colonies that reproduce via a single fertile, egg-laying queen. Like humans, they are warm-blooded, omnivorous and breathe oxygen, yet they are eyeless and communicate and rationalise their surroundings through a form of natural sonar.
It is sourced from Aliens Adventure Game (1991), Sea of Sorrows (2014), and River of Pain (2014).