During the pre flight inspection, usually the copilot checks many parts of the aircraft. Which are the parts and Why do they check them?
Many things are checked on a walkaround. The pilot-in-command should do the walkaround, not the co-pilot. The main classes of items are as follows:
structural integrity; the aircraft must be sound and undamaged and ready to fly
sensors; sensors such as antennas and gauges should be undamaged and not have anything blocking them or adhering to them
clean flight surfaces; the control surfaces and flight surfaces must be smooth and clean; for example, the wings must not have ice on them
no leaks; hydraulic fluid, oil, fuel, and other critical liquids should not be leaking out of the aircraft
chocks, control locks and covers removed; all chocks, protective coverings and locks should be removed before flight and stowed
correct tire inflation; wear on the tires should not be excessive and the tires should be inflated properly
fasteners present and correct; all bolts, shims, nuts and other fasteners should be present, undamaged, and correctly installed
hatches and fairings secure; removable inspection plates and any other small doors, fairings and cowlings should be in place and secure.
lights ok; the lights on the aircraft should operate correctly and their lenses should have no cracks or other defects
blades ok; the propeller or turbine blades (as the case may be) should be perfect with no cracks or chips
Overlooking any of these sorts of items can cause (and has caused) serious accidents or incidents in flight or during takeoff and landing.
The following is an (incomplete) list of checks carried out:
Engine - for any damage to blade (due to birdstrike etc.)
Tires - For any damage
Oil level indicator
Wings- Leading and trailing edges and tips for any damage
Any oil leaks
Rear fuselage- tail skid area
Engine and fuselage- Any open cowling, exterior damage etc.
Probes- Any damage, ice etc.
Any other suspected damage.
In case of helicopters, condition of rotors for any damage.
The other answer gives what is checked, but doesn't really touch on the "why".
Here are a few examples:
Birgenair 301 - Crashed shortly after take-off because it is believed a pitot tube was blocked by a wasps nest.
Aeroperu 603 - Crashed shortly after take-off because some tape was covering the static ports left there because of maintenance work.
Gulfstream IV - Crashed during take-off because the gust locks were not deactivated (Gulfstream had some role to play in this because the gust lock system is supposed to limit throttle movement but was not operating properly).
Twin Otter crashed just after take-off because the control lock was still in place.
Medical Flight crashed after take-off when it was refueled with JET-A instead of 100LL.
As a pilot, there is little you can do in the air when something goes really wrong, but there is a lot you can do on the ground to make sure that the aircraft is worthy to fly. Preflight inspection is one step to a successful flight and preflight doesn't just happen on the tarmac walking around the plane, there are also safety checks that happen inside the cockpit (like verifying the controls are free) and even more before you even see the plane (checking weather, routes, destination airports, NOTAMs, etc).