They are always used for take off and landing.
The trailing edge flaps can be set at different angles, and they are normally deflected much more for landing than at take off.
In an emergency you may be able to land without extending them, but it would be quite tricky with a heavily loaded plane, and it would mean a lot more wear on wheels and brakes. And a much longer roll on the eunway.
If the runway is long enough it might be possible to take off without them, at least with a lightly loaded plane. But I would bet that it is illegal and a crew doing such a stunt would never fly again.
The common name for these things is "high lift devices". Remember that the wing is designed for efficient flight at 500 mph at thinner air at cruising altitude. That corresponds very roughly to 250 mph at sea level air density. This is far too fast for take off and landing. Therefore these high lift devices assist with greatly improved lift with a huge drag penalty. The drag doesn't mean very much because they are extended for only a short time - few minutes at most - and in the dense air at low altitude the engines can produce far more power that at high cruising altitude.
Remember what 737LAME wrote - go and borrow books about aerodynamics. Those high lift devices are much more complecated than described here, and their function - as well as aerodynamics in genereal - is facinating.
When studying high lift devices you will soon learn that they are nothing invented by mankind. Birds used them millions of years before the first man stood upright on his two legs.
Best regards, Preben Norholm
Always keep your number of landings equal to your number of take-offs