When recording on film, the crew cannot review the footage straight away, as film needs developing. In addition, the director cannot actually see what the camera is recording, as it is recording onto film. How do they get around this? How can they see if everything is correctly framed, in focus, etc, when recording on a medium that still requires processing?
Do you mean in this day and age, or back in the day before we had digital cameras?
You can get a tap from the camera these days, but you still can't see what was actually captured to film until it gets back from the lab.
In short, you need a DoP who knows film; it's too expensive for amateur guessers.
There is a system known as a Video Tap or Video Assist using a beam splitter to separate out a tap to a modern CCD system [like in a phone or DSLR camera] which can live feed to on-set monitoring, so the director and crew can see the feed as it is shot...
These days, of course, live playback is the norm - every department needs a feed - also a single-camera shoot is a rarity, 2, 3 or 4 is quite normal. Every camera to the director & the DoP, one camera to each focus puller, a composite to costume, makeup...
Video assist is a system used in filmmaking which allows filmmakers to view a video version of a take immediately after it is filmed.
Originally a small device, called the video tap, was installed inside a movie camera that allows (with the addition of a monitor) the director to see approximately the same view as the camera operator, and thus ensure that the film is being shot and framed as desired. This is done by using a small charge-coupled device (CCD) (similar to ones in consumer camcorders) inside the viewfinder. On modern film cameras, the assist is fed off a beam splitter, which splits the beam between the optical viewfinder and the video tap. The light enters through the lens, and hits the rotating mirror shutter, which bounces the light to the horizontal ground glass. The beam splitter is directly over the ground glass and turns the light again 90 degrees, and projects it onto the chip of the video assist camera—through its own lens system. The chip, together with its electronics, lens system and mounting hardware, is the video tap, and was commonly called video assist until the video assist industry grew large.
The rest of that article then actually goes into how video playback is distributed to the various crew departments that need to see the footage as it is shot on any large shoot, not specifically that for a film/video splitter.
Back in the day, there was no chance of immediate playback - nor could the director see what the cinematographer shot. They had to trust one another. The director had to look for the performance, the cinematographer for focus, lighting, framing etc.
That's probably why when you see old stills of shoots, the director is sitting really close to camera, to get an idea of what the shot would be, without actually being able to see it.
These days, of course, they can sit somewhere away from the kerfuffle and watch it on a monitor.
 DoP = Director of Photography also referred to as DP [which I think is US, DoP is UK] or the 'old' term, Cinematographer.
These days, the DoP rarely actually holds a camera, the camera-op does that, assisted by up to 3 others, focus-puller, clapper & grips. (Other more politically-correct designations are available ;)
 I actually disagree with this point from Wikipedia - the whole idea is you can see it as it's shot, not have to wait until afterwards. Once you have the digital tap, you can handle it in exactly the same way as a direct shoot to digital.
Prior research and links for interest...
Cameron: Specifically for Westworld I had Arricam Lites retrofitted for HD taps. Quite frankly, if directors and everybody else saw the video taps that we used to use on film cameras they’d flip out. Jonathan [Nolan] is actually very experienced with shooting film and looking at classic video taps, but still I thought it was important to establish a set protocol to get a better image for both Jonathan and various department heads so that they could see what we were doing. We dug up a bunch of video taps from around the country and put them on the Arricam Lites. The problem was we were shooting in Santa Clarita, California in the middle of the summer at high noon, when it’s about 195 degrees. So we tended to have a few problems with the HD taps. Just picture these Arricams with ice packs wrapped around the video taps all the time to keep the temperature down. (laughs)
While I felt confident that I could do it without playback for myself, I agreed I was already walking a tightrope, and what maniac doing a circus act for the first time would refuse a safety net? So we got a video tap for the camera that captured the image going through the gate and recorded it into a DV recorder with a hard drive. Our monitor was a tiny, fuzzy screen half the size of an iPhone, and the only reason to watch was to see what was in frame and to check performances. Sometimes the hard drive would fail, and we would have to record video of the video tap on an iPhone for playback.
Before the video assist that the fine answer from @Tetsujin described, and perhaps even afterwards until all-digital production, you had the Dailies/Rushes where each day's raw film was developed.
From Wikipedia: Dailies
Dailies, in filmmaking, are the raw, unedited footage shot during the making of a motion picture. They are so called because usually at the end of each day, that day's footage is developed, synced to sound, and printed on film in a batch (or telecined onto video tape or disk) for viewing the next day by the director, some members of the film crew, and some actors. Dailies serve as an indication of how the filming and the actors' performances are progressing.1 However, the term can be used to refer to any raw footage, regardless of when it is developed or printed.
Another way to describe film dailies is "the first positive prints made by the laboratory from the negative photographed on the previous day".
Dailies may no longer be needed (or maybe they are, especially for cgi effects, see comments below), but there are still daily "film"-related processes that happen.
When using a video camera or digital motion picture camera, the image and sound are often recorded simultaneously to video tape or hard disk in a format that can be immediately viewed on a monitor, eliminating the need to undergo a conversion process to create dailies for viewing. The footage recorded each day will still usually go through a daily process to create a second copy for protection and create multiple copies on DVD or other media for viewing by producers or other people not on set.