Depending on your desired ship, there are several different ways to obtain them to order as you see fit. They are described below. You may wish to equip drones to some ships (Capital Ships and Medium Miners). This can be done when ships are purchased and in other ways, which are all explained on the Drone page.
Refer to the articles about Boarding Ships for more info.
Simply put, you order your boarding team to try to overrun and take control of the ship. Note that it will reduce your reputation with <target> faction (as well as any faction nearby that is allied to <target>) to do so.
NOTE: This (mostly) applies to S/M ships!
When in dogfights with S/M ships, sometimes the Pilot in that ship bails out in an Escape Pod. The Small ship then becomes "Neutral" and does not function at all. The escape pod remains hostile and can be shot up with no penalty (unlike previous X games such as X3: Albion Prelude).
You will need to recruit a Pilot in order to claim the S/M ship. Once done, return to where your <target> ship is & double-click on it. Click "Claim" and wait for your Pilot to travel in a pod from your ship to the target (slow) and then claim it.
There are random ownerless ships scattering in various locations in the entire X: Rebirth Universe. If you happen upon a capship (L/XL) rather than a S/M ship you'll need at least a Captain to perform a similar feat (Pilots cannot work on capships; Captains cannot work on S/M ships).