I have flight where I need to move from airplane to another. Do I need to check out my luggage and then check it in again, or will it be automatically transferred to another plane?
That depends. If you booked the flights together at the same time, luggage will usually be "checked through" to your final destination. If you booked them separately, especially if it's different airlines, you'll probably have to pick up your luggage and check it in again.
In any case, the airline employee at your original checkin can tell you whether your luggage is checked through - they'll often ask you themselves to confirm whether you want that.
Usually, an airline checks luggage to the final desination when transfers occurs. On a same booking, as already said, there are many chances your luggage goes on through to the next plane.
Separate reservations on different airlines
Airport change during a transfer : you will need to bring the luggage with you during the required ground transportation
Transfer from an international inbound flight to a domestic outbound flight : since this is your entry point into the country, you will have to pick your bags, clear the customs then re-check them right after. This is the case at least in the USA, and depends on the countries. In Europe Schengen space, it did not happen when flying MIA -> FRA -> CDG although FRA is the entry point into Schengen space.