The tech crew is very important to a stage production. They are the ones responsibles for ensuring that sets are built, lights are aimed properly, sound occurs at the proper times, all the costumes are fitted properly, and that everything goes off without a hitch.
The stage manager works directly under the director and it is their job to call any actors who are late, take line notes to ensure that the actors are learning their parts correctly, and be ready to fill in for missing parts at a moments notice. The stage manager is very important to keeping the production running on time and working with the director to get through any issues that pop up.
The lighting director is also very important to keeping the show running smoothly because they are the one that insures that every scene is properly lit for the setting. This takes alot of hard work to get right due to the number of lights invovled in making one scene look correct.
The sound director's job is twofold. He first must ensure that any sounds that need to occur in a scene happen at just the right moment. If they are off by half a second then the scene can derail and lose cohesion. Secondly they are in charge of any music that occurs during the show. Wether it be before the show starts, during scene changes, or at the end they are the ones who keeps that going.
The tech assintants are the minions of the rest of the tech crew. Any little job that pops up that the main techies need done is often delagated to these people. Not to say that they aren't important, in fact the show might not run properly without them. These are the people who help run wires, set lighting trees, help adjust the angle of the lights while thew lighting director changes setting, and other jobs in a similar vein. They arevery important to the show and should not be forgotten.
The costume mistress is in charge of everyones costuming and prop needs. Their job is one of the hardest due to the need to make changes to each individual costume in order to ensure that they all fit properly. This can take weeks of measuring and fitting for each outfit with multiple outfits being worked on at a time. They also have to find appropriate props to go with each character based on the scene. This requires an understanding of the play, the individual scene, and how the character fits in the scene.