The crew people with whom you're most likely to interact on set...

The director of photography tells the story emotionally through visual means. Lighting is a big part of this, and so are camera placement, camera moves, lens choice (usually in collaboration with the director).
CAMERA OPERATOR (often known simply as "Operator")
The camera operator pans and tilts the camera to follow the action and will be able to tell you what the framing is, how much range of motion you have within that frame, etc. If you have a particular action in a scene, it's the camera operator's responsibility to capture that action (e.g., you are sitting and you stand up), so it's wise to find a way to make it clear to the camera operator when and how fast this will occur (on a particular line, or if you have a subtle signal, etc.) so that they can follow you properly and not lose you during your movement. (The Director of Photography will also quite often operate the camera.)
FIRST CAMERA ASSISTANT (also "First AC" or "Focus Puller")
In charge of keeping the image in focus, often without actually being able to see if it's in focus. Traditionally, the 1st AC will measure the distance to the subject, then set that distance on the lens. He will then adjust that setting on the lens as the subject moves closer or farther from the lens. However, it is becoming increasingly common that focus pullers now use a video monitor to see the image and keep it in focus.
SECOND CAMERA ASSISTANT (or "2nd AC"; sometimes also "Clapper/loader," mainly in England)
Holds the slate in front of the camera and claps the sticks at the beginning of each take. Sometimes, especially on a close-up, the slate will be right in front of your face. Experienced 2nd ACs will call "soft sticks" to the sound mixer, indicating that he or she will clap the sticks softly so as not to disrupt the actor (the mixer will record the clap accordingly), but inexperienced 2nd's often do not know to do this and may bang the sticks loudly right under your nose. The slate will sometimes be clapped at the end of a take instead of at the beginning ("tail sticks" or "tail slate") for various reasons, so you can't always rely on that clap as the signal to get ready-ready. (Also: nowadays, shooting digitally means that it's cheap to keep rolling the camera, which means directors will often want to go again or go back to the top or back up and pick up a line or section of the scene without cutting, because cutting and re-setting can disrupt the momentum of the moment. Be prepared for this to happen, and if you need a moment to get back to a particular place, take that moment.)
Head of the lighting department. Takes instruction from the Director of Photography about the desired look of each particular shot, and issues orders to the electrical crew about how to achieve this look.
Sets the lights as instructed by the Gaffer, and also performs any lighting cues during takes. Sometimes, you may have to interact with lighting during a take (turning on light switches, lamps, flashlights, etc.), and you may work with the electrical department to facilitate that. For example, a lamp or flashlight may have a power cord attached, which will be hidden down your sleeve and then run down your pants leg. An electrician will look after this cord as you make your way along a dark corrider.
Head of the grip department.
Grips are perhaps a little like stage hands or roadies. Among a grips duties: shaping the light--either by setting flags to keep light from hitting things it shouldn't, putting up fabrics to soften the light, etc.--as well as moving certain things around set, including the dolly (a wheeled support for the camera that enables it to move with the actors during a take. The grip who pushes the dolly during the take is called the...
Like the camera operator, the dolly grip needs to move the camera at prescribed times, so you should make him or her aware of what you're going to do and when you're going to do it, so that they can follow your actions properly.
Runs the set, like a traffic cop. In charge of scheduling each day's work. In charge of knowing what needs doing to achieve the shot in question, and making sure everyone is doing what they're supposed to. In charge of keeping the shoot on schedule, and for keeping the set safe. Rarely leaves the set, is almost always at the director's side. Will attend blocking while the shots are figured out, will stay with the set while it is being lit and readied for the shot, then runs the set during shooting: after making sure the crew is ready to shoot, calls for the actors; once the actors are in place, calls for quiet, calls for sound to roll, calls for camera to roll, then often calls "Action" (depending on the director's preference), and calls "Cut" after getting the signal from the director to do so. After the shot is completed successfully, facilitates moving on to the next shot. Often loud, but does not necessarily need to be. Needs to be able to make himself/herself heard and understood.
SECOND ASSISTANT DIRECTOR (or "Second AD" or "Second")
The First AD's lead assistant. Because the First AD can rarely leave the set, the Second is in charge of any off-set management, including making sure the actors are in hair/makeup/wardrobe and are being readied on schedule. The Second is often the person sent to fetch the actors when it's time to shoot.
PRODUCTION MANAGER (also "UPM", for "Unit Production Manager")
Runs the production. In charge of where everything needed for shooting will be placed (trucks, etc.) and how it will get there. All departments report to PM for logistical issues.
Works for the production manager to keep the real world running.
The go-fers of the set. Usually relatively inexperienced, they do whatever they can to support production. They are also often assigned to help individual departments.
During a take, the sound is recorded separately from picture, and the sound mixer is the person who records it. The mixer is in charge of choosing which microphone to use, and how to position it: shotgun microphone on a boom? Lavalier ("lav") body mic? Both? In charge of the volume level of recording: louder when the actor speaks quietly, quieter when the actor is loud. Needs to capture the dialogue cleanly and clearly. Will sometimes need to ask for more volume from an actor (e.g., if the location is noisy, etc.). Will also need to know if the actor is going to suddenly get very loud, so he can adjust his recording properly as it happens. For example, if an actor yells suddenly and unexpectedly, it is likely that the sound of that yell will be distorted on the recording, and thus unusable. The Sound Mixer will also have to sometimes let an actor know if there is a technical issue (e.g., if the actor is rustling a paper bag while unpacking groceries and causing too much noise, or if a floorboard is loose and squeaks when an actor walks or shifts their weight). Usually, the mixer will go to the AD with these concerns, who will then figure out where to go for the solution. But sometimes, the mixer will go straight to the actor with these adjustments. Also, body mics (lavs) tend to shift around or rub up against cloth or hairy chests, etc., and will often need to be re-adjusted between takes.
Works with the sound mixer. The boom operator is in charge of pointing a shotgun microphone at the actors; the microphone is generally at the end of a boom pole. The shotgun microphone is highly directional (picks up sound only from the place where it's pointing), it has to be pointed at whoever is speaking, so the boom operator must be familiar with the script and must have a very good sense of which actor will speak when. This is critical during a two-person master shot, but we will often also use an actor's off-camera dialogue, so that should be recorded properly as well. Also, the boom operator is generally in charge of placing any body microphones and transmitter packs on the actor (and replacing the battery in the transmitter as needed, etc.)
SCRIPT SUPERVISOR (sometimes called "SCRIPTY")
In charge of continuity. Will keep a record of what was said and done by whom and where for each take. (For example, "Actor A picks up pencil with left hand on the 'hello' line."). Will make a note of issues during takes (eg. "lit the cigarette on a different line in Take 4..."), then let the director know of such issues, so the director can decide how to proceed (i.e., re-shoot the shot or move on). Depending on director preference, the script supervisor might also directly approach the actor with a continuity issue. Will also provide lines for actors as needed.
In change of any physical items actually handled by the actors. Will place the props on set for the scene, and will give any needed hand props to the actor. Will instruct actor on special use of any hand props as needed.
In charge of dressing the set. May sometimes consult with an actor to design a space in which the actor lives and/or works.
Will style the actor's hair in the appropriate fashion, and then be available on-set during shooting to make adjustments as needed. (A common problem is the "fly-away" hair that becomes very noticeable to camera due to lights placed behind the actor (ie. "backlighting".)
Will apply the appropriate makeup to the actor, and then be available on-set during shooting to make adjustments as needed.
Will supply the actor with the appropriate costume for the scene being shot, and be available on-set during shooting to make needed adjustments and repairs.