NEF is Nikon's RAW format
CR2 is Canon's RAW format
NEFs and CR2s can be processed in Adobe Lightroom. However you may need need to update Lightroom if the version you have came out before the T5i. You would have the same problem with a recent Nikon camera. Everytime Nikon and Canon bring out a new camera model, they change their RAW format - and updates are required for your RAW processing software.
I process T5i CR2 files in Adobe Lightroom 5. And it works just fine.
In the meantime, you can download Adobe's DNG converter for free - from the Adobe site, and convert the CR2s to DNG, and then process the DNGs in Lightroom as you would any RAW image.
Also your T5i was supplied with Canon's Digital Photo Professional (DPP), which can process your CR2 images. It's on the disk that came in the box.
There are also other RAW processing applications which will work with T5i CR2s. Including RawTherapee, GIMP with the UFRAW plugin, and Photivo. All these are free.
You will not lose any quality if you use any of these free methods I have listed.