Pilots usually use Flaps 1 when the weather is good for flying and if the runway used is longer than needed. Flaps 2 is used usually when headwinds are not strong enough or the runway in use is not that long for a takeoff run.
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Touching down with full flaps gives you the lowest stalling speed. It also gives you an additional benefit: maximum drag so you will decelerate as rapidly as possible with the power off, getting you down through that red zone of poor control, rapidly.
Full flaps increase drag more than they increase lift and is used to slow the aircraft for landing. Taking off with 10 to 20 degrees of flaps increases lift for takeoff without incurring a lot of drag, which you don't want for taking off.
Most airlines IRL use Flaps 30 as the standard. It would go something like (1/5/15/25/30) 40 is used when there are higher wind conditions of the sort to try and stablize the aircraft on approach.
So a Flap 3 landing would be a landing with the flap handle in the Flaps 3 position, one short of full extension. Why would a pilot want to do that? The ideal landing speed would be a little higher than with full flaps, which might be advisable if the winds are gusty and the runway length is not a constraining factor.
Most of the time, landing flap will be determined by the landing performance required, flap 30 is nearly always desirable unless we need the lower approach speed, or if increased forward visibility is required such as during low vis ops. Performance is the determining factor.