Director Ridley Scott deleted a couple of scenes from the final version of Alien. These can be looked at down below, by clicking on the pictures. To my knowledge almost all deletions have had an artistic or budgetairy reason and are not due to any influence by Fox.

(Thanks to the Aliensaga website: http://atreides.simplenet.com/aliensaga)

You are advised to install Intel Indeo 3.05  and a proper mediaplayer (Netscape/Microsoft).
 

The crew of the ship listen to the warningsignal of the derelic ship	Ripley informs about the status of the repairs.
Lambert is mad at Ripley as she refused to open the airlock after the visit to the derelic ship on LV426.	Lambert visits the lab and asks Ash about his progress on Kane's situation.
After the "birth" and escape of the chestbuster alien the group is discussing following actions.	The alien attacks Brett when he is looking for Jones (the cat).
Parker tries to catch the alien in the airlock. This scene was only filmed for this part and not finished. It would also have contained a different dying of Lambert (by decompression) and Ash alarming the alien.	Ripley asks Lambert about her relationship with Ash.
The alien crawls towards Lambert before it kills her. This part was cut out scene was cut out of the movie, because it showed to much of the xenomorph. No sound.	Ripley runs into the cocoon of the alien and finds Dallas alive. These scene was deleted, because it wouldn't do good to the pace to the end of the movie.