There is faster than light travel in the Alien universe franchise. It is demonstrated by the relatively short time (3 weeks) it takes for the Marines to arrive at LV-426, Zeta 2 Reticuli system (second moon of Zeta 2 Reticuli IV).

• This star system is 33 light years from Earth, so their ability to reach it in such a short time indicates a well-developed FTL system, whose specifics are undefined. (Such information is not necessary to the story so it's overlooked.)

Click to enlarge map of the Alien Universe ^[1].

• On starships where travel between distant interstellar locations would take centuries due to slower-than-light travel (i.e. generation-ships or anything made by humanity at our current state of technological development), it makes sense to put the crew into suspended animation since it will take thousands of years to reach their destination. Even our closest stellar neighbor, Alpha Centauri, which is only four light years away would take 25,000 years at our best possible speeds!

• The Alien universe possesses faster-than-light travel so their reasons for suspended animation are likely based around the second - and equally important - reason for suspended animation after time delay: the important issue of resource management.

• Living, moving, active crew require food, oxygen and water to go about their business of being human. It would appear, travel between stars requires very little human interaction except the initiation of the process so it makes more sense to reduce the activity of the crew because it reduces their need for resources.

• Physically, the suspended animation process seems to have no long-lasting ill effects (witness Ripley and Jonesy's decades-long sleep and revival) so it makes sense the humans of that universe would spend their time asleep to reduce the need for storage space for water, food and air as well as entertainment.

• It is very likely they have technology to promote the efficient recycling of those resources, but since no replication technology appeared to be available, the easiest best way to have food on hand is to NOT eat it in the first place.

For those of you curious about the ships and technology of the Alien Universe, The Alien Universe Timeline ^[2] offers very interesting materials based on canonical resources.

I recall the Aliens: Colonial Marines Technical Manual ^[1] introducing the concept of negative (or inverse?) time dilation for FTL travel.

That is, much as the journey-time experienced by the crew of a ship traveling close-to, but below the speed of light, is less than that experienced by a stationary observer, the crew on a FTL journey would experience a longer journey than perceived by a stationary observer.

So, in the three weeks journey to LV-426 above, the crew would have experienced a considerably longer travel time, and therefore required cold-sleep to avoid wasting a large portion of their lifespan on travel.

According to the website ^[1] the Weyland corporation had two patents of interest:

1. Weyland FTL Lifeboat: Exclusively designed for FTL craft of a suitable size, this emergency sustaining pod is an embedded safety module comprised of 7 luxury-appointed rooms, complete with independent navigation and piloting controls. Reclaiming decontamination system allows one occupant 50 years of breathable air with or without hypersleep.

2. Heliades: As the first-ever FTL space exploration vehicle, Heliades revolutionized the known galaxy. Dual FTL engines are driven by a 2.1 terrawatt RL fusion reactor. The ability to provide molecularly stable transport for ship components and human crew over light-years of space gives Weyland explorers, scientists and planetary engineers access to regions never before imagined. ICC-certified and compliant, our newest flagship model has a payload capacity of up to 9 megatonnes and a redesigned airframe and features a medical bay, lab facilities, and over a dozen HES-compatible chambers.

In May 2032 (only a few years from now!) "Weyland scientists discover the inverse relationship between velocity and the flow of time making the long sought-after concept of faster than light travel a reality. The search for practical application begins." While by January 2045 "Weyland Corp introduces the first FTL-capable SEV (space exploration vehicle)."

In Alien (1979), the "Nostromo" is in the Zeta 2 Reticuli system which is 39 light years from Earth and it is said that the journey will still last 10 months. Which implies a speed of about 47 times the speed of light.

From transcriptions:

RIPLEY: Calling Antarctica traffic control. Do you read me, Antarctica? Over. (x2)

LAMBERT: I've found it, just short of Zeta II Reticuli. We haven't reached the outer rim yet.

...

DALLAS: How far to Earth.

LAMBERT: Ten months.

Edit: This is the only ANSWER, so far that highlights the presence of FTL travel as internal evidence already in the first movie (i.e. "Alien",1979).

In Alien 3 when the warden was telling his prisoners (rumor-control was how he put it, IIRC) that Ripley had crash-landed on the penal colony, he said a ship had been sent to pick her up (I don't remember the ship's exact ETA but I seem to recall it would get there in weeks) so unless it came from somewhere in the same solar system, it probably would need some form of FTL to reach the colony in that span of time.

For the USS Sulaco to have gone from Earth (Sol System) to LV-426 (Zeta 2 Reticuli) they would have had to been traveling at 13 times the speed of light. Since they got there in 3 weeks and Zeta 2 Reticuli is approximately 39 light years away.

As per Thaddeus Howze's answer, ships are definitely capable of FTL travel in the Alien universe. But I don't think the stasis pods are there for dealing with time dilation.

First, modern subs carry a year's worth of food and they are cramped and have limited space. Water is another issue but can be recycled easily as with the International Space Station. There is another reason for the stasis.

Keep in mind that FTL travel often means the space the ship is on is traveling faster than the speed of light, not the ship itself. So no time dilation would be experienced by the crew. See Alcubierre drive ^[1]. The limiting factor for FTL travel would absolutely be the availability of exotic matter needed.

It is more than likely that in the Alien universe this matter is also extremely rare and extremely expensive to produce. Therefore​ some missions of great importance (the marines missions to investigate the lost colony) could be authorized to use FTL travel. Cargo missions would be relegated to using sub light propulsion and the necessity of stasis becomes paramount.