Canon 60D is an APS-C format DSLR that can fully use both EF and EF-S lenses. But because of the smaller sensor format, you will likely want different focal lengths than someone using a full frame camera like a 5D Mark IV.
While certainly not a hard and fast rule, a "portrait" lens is usually a short telephoto. There are MANY of those available for 60D.
50mm lenses make for a good "short portrait" lens (much like an 85mm on full frame). There are the affordable Canon EF 50mm f/1.8 STM, the mid-grade EF 50mm f/1.4 USM and the premium EF 50mm f/1.2L.
On a 60D, 85mm make for a good "long portrait" lens (much like 135mm on full frame). There are plenty of these to choose among, too. Canon EF 85mm f/1.8 USM is a mid-grade lens, while EF 85mm f/1.4L IS USM and EF 85mm f/1.2L II USM are a couple of more premium option.
The shorter lens may be better if you have limited work space, such as a small studio setting. It also may be better for head and torso portraits, or for couples. The longer focal length might be preferable for more candid shots taken from a greater distance, or for tighter head shots, so long as you have sufficient working space.
Lenses in this range of focal lengths are used because of the way they render peoples' faces. Shorter focal lengths tend to exaggerate facial features, while longer focal lengths tend to compress them.
This is not to say that other focal lengths can't be used. For example a 35mm or 28mm might be wanted for full length portraits, especially couples like a bride and groom. The wider lens also may be useful for environmental portraits, which show a lot more of the person's surroundings, such as their home or workplace. The key thing with shorter focal lengths is to take care not to get too close to the subject and not to compose with them too near the edge of the image, which can cause strong exaggeration and distortion.
Longer focal lengths certainly can be used, too. Sometimes compression is desirable. For example a lot of fashion photographers like to work with longer focal lengths. On a 60D this would be 135mm and 200mm lenses... or possibly even longer. Canon's EF 135mm f/2L USM lens has near legendary reputation, though is a long lens requiring a lot of working space on an APS-C 60D.
You've probably noticed that I haven't listed any zooms. There are a couple reasons for that. One is that few zooms offer any larger than f/2.8 aperture, and even those tend to be pricey. Prime lenses can be a lot smaller (less intimidating to subjects), more affordable and can offer larger lens apertures. Often portraits are done with large apertures in order to strongly blur the background, to make the subject stand out. This is especially true for location work, where you have less control over the background than you do in a studio. Some types of portraits also go for a "dreamy" look, which the large aperture helps achieve.
But there are also instances where a zoom can be valuable. Kids and pet portraits often require very fast shooting where the versatility of a zoom can be quite helpful. Indoors I might use a Canon EF 24-70mm f/2.8L USM (either version), but outdoors with kids and pets an EF 70-200mm f/2.8L (any version) may be a better choice. A very affordable option is the Canon EF 28-135mm f/3.5-5.6 IS USM lens. Dating back to the days of film and one of the first lenses to feature image stabilization, it's sort of a "sleeper" that was often sold as a kit lens with various cameras. As a result, even though it's now discontinued there are A LOT of them around and they are pretty inexpensive... Yet the are very good performers! The 28-135mm has very good image quality, fast autofocus, helpful stabilization and a nice zoom range for portraiture. I've used several copies of the lens myself, over the years. I also know of at least two professional portrait photographers who swear by it! Nothing fancy... just a very capable lens that does the job well at minimal cost.
Again, there are no hard and fast rules when it comes to portraiture. In addition to all the above there are 45mm, 90mm, 100mm and 105mm lenses that might serve well for portraiture. For example Canon has made an excellent mid-grade EF 100mm f/2 USM in the past (another discontinued lens).
There also are macro lenses that some people choose to serve double duty... both for portraiture and for close-up/macro work. Canon's EF-S 60mm f/2.8 USM and EF 100mm f/2.8 USM are a couple mid-grade options... while the EF 100mm f/2.8L IS USM is a more premium choice. One bit of caution... macro lenses tend to be very sharp, capturing every little detail. Not all portrait subjects appreciate that!
Even Tilt-Shift lenses (also called Perspective Control) like Canon's TS-E 45mm f/2.8, TS-E 50mm f/2.8 and TS-E 90mm f/2.8 (both versions) also can be used for portraiture, although they are manual focus lenses so will be best used with stationary subjects in a more deliberate setting. Tilt-shift lenses offer some interesting effects with the plane of focus.
All the following were shot using various lenses on APS-C format cameras like your 60D...
Canon EF 50mm f/1.4 USM lens at f/2
Canon EF 85mm f/1.8 USM lens at f/2
Canon EF 135mm f/2L USM lens at f/4.5
Canon EF 300mm f/4L IS USM lens at f/4
Canon EF 500mm f/4L IS USM + Canon EF 1.4X II (700mm combo) at f/5.6
Canon EF 24-70mm f/2.8L USM lens at 30mm and f/11
Canon EF 24-70mm f/2.8L USM lens at 58mm and f/11
Canon EF 28-135mm f/3.5-5.6 IS USM lens at 117mm and f/7.1
Canon EF 70-200mm f/2.8L IS USM lens at 160mm and f/5.6
Some of the lenses mentioned above are discontinued, so will only be available on the used market. Also check for refurbished lenses at the Canon website. Those are little different from buying new (same warranty), but will save you a few $.
