You don't need radio triggers to fire the YN-565EX remotely from the D610.
The D610's pop-up flash can actually remotely fire the 565EX in three of its optical slave modes without any additional gear.
YN-565EX's Optical Slave Modes
S1, S2 "dumb" optical slave modes
If the D610's pop-up flash is in Manual mode, the 565EX's S1 mode (which is similar to Nikon's SU-4 mode) can be used as a "dumb" optical slaving method to fire the 565EX. You don't have any control over the remote flash other than firing in sync, but it will work. Similarly, with the pop-up flash in iTTL, you can use the S2 optical slave mode--this will ignore the pre-flash for the iTTL metering. These two modes will work for any type of flash burst--even that from a P&S camera or a studio strobe, so they're an easy way to integrate the flash into a lighting setup.
SL (CLS slave) Mode
If the D610's pop-up flash is put into CLS Commander mode, the YN-565EX's SL slave mode makes it a CLS slave. This can only be used in concert with a Nikon CLS commander unit, but will give you wireless iTTL, FP (HSS) flash, and remote power control over the flash. (Weirdly, it will also work with Canon's wireless eTTL system, if you have a friend who shoots Canon).
These are all optical methods for triggering the flash off-camera, and are probably your best bet for getting started with off-camera flash. Optical slaving's main weaknesses are line-of-sight and reliability in bright light, but indoors in studio-type conditions with a lot of bounce surfaces, and lower light levels, optical slaving systems can work very well. It's only when you shoot with off-camera flash outdoors on location in bright sunlight that you may wish to move to radio triggers to eliminate the line-of-sight and range restrictions of optical slaving.
Adding Radio Triggers
With radio triggers, you do indeed need two units: a transmitter to be on camera to act as the commander/master; and a receiver on the flash to slave the remote flash to the camera. Most of the lower-cost triggers today can connect directly to the camera hotshoe and the foot of the flash, but some of the higher-end triggers may only connect to the camera hotshoe or a PC sync port. The YN-565EX has a PC sync port, however, so as long as you've got the right cable to hook the trigger to the flash, you should be good to go.
With radio slaves, there are two main kinds: manual-only and TTL-capable. Manual radio triggers tend to be cheaper, which makes them more attractive, but just as with the S1/S2 optical slave modes, all you can tell the remote flash to do is fire. You'll have to set the zoom/power level on the flash itself, you won't have TTL, and your shutter speeds will be limited to at or below your D610's sync speed (1/200s).
For examples of this type of trigger, see the Flash Havoc blog's Manual Trigger page. Radio triggers, both manual and TTL, are constantly evolving and changing and current models are always turning over.
TTL-capable triggers can communicate most of the hotshoe signalling protocol, so whatever you can do with the flash on the camera hotshoe, you can do with it remotely. The two features most off-camera flash shooters like these triggers for are high-speed sync (i.e., no longer having the shutter speed limited to x-sync speed) and remote power control.
For a good list of current TTL triggers, see the Flash Havoc TTL & Remote Manual Radio Triggers page.