How did the crew know where this character went, and where the Alien was going?

Question

In Alien, there is a scene in which

Brett is killed by the Alien while searching for Jones the cat.

After this happens, the other members of the crew know this happened, and apparently know where it happened, saying that the Alien must have gone through the air ducts. There don't seem to be many surveillance systems on board, as, when the Alien disappears from Kane's face in the medical ward, they don't have any idea where it went. Then, when it is being chased through the air ducts, they have a small display of where it is. How are they tracking the Alien, and how do they know what happened to Brett?

Answer 1

From the screenplay:

How are they tracking the alien

Ash has constructed motion detectors:

                             LAMBERT
              Now if we could only find it.

    Ash picks up a portable unit.

                             ASH
              I've taken care of that...tracking
              device.  You set it to search for
              a moving object...It hasn't much
              range but when you get within a
              certain distance it starts beeping.

    Ripley takes the tracker from Ash's hand.

                             RIPLEY
              What's it key on.

                             ASH
              Micro changes in air density.
              Keep it pointed ahead of you.

How do they know what happened to Kane

I'm not sure if you mean Kane, or one of the other characters. Kane is the one who has the facehugger attached to him. His fate is witnessed by the rest of the crew when

the alien bursts out of his chest while he is seated in the mess hall with them all.

UPDATE: How do they know what happened to Brett

Brett, Parker, Ripley and Dallas try and capture the alien using a net, but end up startling the ship's cat Jonesy. Concerned that the cat wandering around might confuse their motion tracker, Brett heads after the cat to catch it. Then, according to the screenplay..

                             BRETT
              Here kitty...Come on Jones.

    Brett reaches for Jones.
    Jones hisses.
    An arm reaches for Brett.
    The Alien.
    Now seven feet tall.
    Hanging from the undercarriage strut in reverse position.
    Grabs Brett and swings up into darkness.
    Brett screams.
    To no avail...
    In the doorway Ripley and Parker.
    They witness the horror.

In the scene as actually shot (from the director's cut), Ripley and Parker arrive moments after the alien and Brett have disappeared but it's clear to them that the alien has taken Brett.

Answer 2

The trackers could possibly network to build a map of the duct system. Also Ripley gets a false positive detecting Jones through an airlock door, so Ash was likely not giving the full explanation.

See 11. ALIEN Frequently asked questions (Movies: Alien):

Q: When the crew first sets out to search the ship for the alien:

• Ash has made a "detection" units (motion tracker)
• Ripley asks "how do they work?"
• Ash VERY hesitantly replies "micro changes in air density"
• when searching, Ripley detects something on the OTHER side of an airtight door (after all, they ARE in a spaceship) which turns out to be Jones.
• Ripley makes the comment "micro changes in air density my ass" This point never gets raised again in the film, what is its significance?

A: [possibly] Ash's hesitation in explaining the operation of the motion detector was because the "real" mechanism would be difficult to explain and he was assuming that she wouldn't understand him anyway... so he pauses before he finds the words to form an "adequate" but not very detailed description. The tone of Ash's voice in response to Ripley's question was somewhat condescending.
Ripley's later comment, "micro changes in air density, my ass" was a foreshadow to show that Ash was hiding something and that she was onto him (cf: she realizes that he didn't give the full explanation of the motion tracker's operating mechanism)
Taking into account that Ash knew what he was going into, he could easily have thought about the possibility that the creature could escape and hide on the ship. Therefore, he could've "invented" this device long before they needed it.