There are various ways to operate turnouts on your layout.
No remote or automatic operation. Requires the Big Hand in the Sky to operate the turnouts. This can quickly become impractical and tiresome on all but the very smallest layouts.
Not recommended when your layout has plently of intricate detail such as signals, telegraph poles, trees, fencing, etc. Something is bound to get damaged sooner or later.
This is a simple form of automatic operation where the points are spring-loaded to hold them in the closed position. When approaching in the facing direction the train will always take the main route. When approaching in the trailing direction the the wheels of the train force the points over if necessary and then they spring back when the wheels have passed. Two of these can be used to make a simple passing loop where the train will automatically take one or other route through the loop depending on the direction of travel.
These mechanisms don't work so well on the smaller scales (HO and smaller) because lighweight items of rolling stock easily derail if the spring tension is too great.
Mechanical Remote Control
A mechanical link connects a lever at control panel location to the turnout's throwbar. The link can be a metal rod, wire-in-tube, or other mechanical method.
Whilst the cheapest form of remote control this can be tricky to install and maintain and has limited practical use on larger layouts where you still have to walk to the location of the control lever.
Electric Remote Control
Turnout Motors are used to electrically operate turnouts and basic installations just use a toggle switch, pushbutton or stud and probe method to supply current to the motor when needed.
This allows remote control over much greater distances but on large layouts you'll end up with a huge amount of wiring and still have the problem that you have to walk to (or someone else has to be stationed at) the location of the control panel.
If the turnout motors are fitted with accessory switches then one of them can be used to light up indicator lamps or LEDs on the control panel to show the status of the turnout. Of course, this doubles the amount of wiring between the control panel and the turnouts!
Custom Electronic Remote Control
You can be as fancy as you like if you're prepared to build you own electronics, anything from diode matrix route switching to complete replicas of CTC control panels.
If you're thinking of going down this route you should first understand what is available with DCC and computer-based control such as the JMRI project.
DCC Turnout Control
Finally we get to what you probably want to know about - DCC-based turnout control.
DCC turnout control uses stationary decoders to operate the turnout motors. These receive DCC commands from a throttle (via a command station) and send power to the turnout motors when needed.
Each turnout motor is assigned an address which is similar to a DCC locomotive address but not the same.
Using a throttle you first select the turnout address then press a button to throw or close the turnout. Generally speaking the throttle will not know the actual state of the turnout but it should know (via the command station) the last command sent that was sent, and therefore what it thinks the turnout is set to and show it on the display (if it has that feature). The downside to this system is that you have no graphical representation of the turnout state or location and you have to memorize their addresses or have them listed on a diagram somewhere.
Some stationary decoders have the ability to accept additional push-button switch inputs to control the turnouts. Then you can mount these switches on a control panel and have the option to use either the local push-button or your throttle. However this doesn't do much for cutting down the wiring that DCC promises; just like the old way, you still have wires running from a control panel to your turnout motors (they just happen to "pass through" a stationary decoder now).
You need to make sure that a stationary decoder is compatible with the type of turnout motor you want to drive. Many of them are configurable for different types of motors but some are not and some are configurable but only for certain types. So check the manufacturer's instructions.
It is possible to mix and match turnout motors, stationary decoders, and DCC command stations and throttles from the various manufactures as long as the stationary decoders are listed as DCC-compatible. But one note of caution; some manufacturers allow you to control your stationary decoders using their own proprietary control networks (Digitax LocoNet being one example) instead of through the track power or another DCC power bus. If you choose to wire up your stationary decoders like that you will be locking yourself into one manufacturer's products unless you do a lot of rewiring later.