APS-C or "crop" cameras can fully utilize BOTH EF and EF-S lenses.
Full frame cameras REQUIRE full frame-capable lenses (i.e. "EF").
By Canon's own definitions, Canon's L-series lenses:
#1. Are made to pro standards using advanced materials and manufacturing techniques (somewhat subjective criteria).
#2. Must include "exotic" glass of some sort (ED, fluorite, aspherical, etc.)
#3. Must be compatible with all Canon EOS cameras past, present and future.
#3 means that there will NEVER be an L-series EF-S lens, since those lenses only fit and work on crop cameras.
Because all L-series are designed for full frame, they tend to be offered in focal lengths considered most ideal for that format. EF-S lenses can and do offer some focal lengths that some might prefer with a crop camera.
But just because they don't have a red stripe or an L labels does not mean that EF-S lenses don't offer performance similar to L-series... especially in terms of image quality, but also in focus speed and accuracy, image stabilization, flare resistance, close focusing ability and possibly other respects. For example, the EF-S 10-22mm USM, EF-S 17-55/2.8 IS USM and EF-S 15-85mm IS USM are all very well respected for their IQ and performance in all respects, even if not L-series or built to quite the same standards.
EF-S lenses do not need to produce as large an image circle to cover the smaller size camera sensor, so an EF-S lens can be smaller, lighter and less expensive than an EF lens.
In general, EF-S lenses are in the wide angle to "walk-around" zoom range. There's little reason to make EF-S telephotos (one exception is the EF-S 55-250mm, made to be affordable... another is the EF-S 60/2.8 Macro, built as a crop only lens to be extra compact).
"Full frame" lenses... whether L-series or not, can work just fine on crop cameras. It really comes down more to personal preferences about focal lengths, since FF designs are typically produced in different ranges than crop designs, as well as cost, size and weight. For example, you just aren't going to find very affordable FF lenses that would be particularly wide on a crop camera... for example, compare the price of an EF 11-24/4L or EF 14/2.8L II with the size, weight and price of an EF-S 10-18mm STM IS or EF-S 10-22mm USM. Compare the cost of an EF 24-70/2.8 with an EF-S 17-55/2.8.
Now, some people prefer a 17-55 or 18-55 "standard" zoom on crop. Personally I like a 24-70 or 28-135 just fine on my crop cameras (and have 12-24 and 10-22mm lenses if and when I need wider).
In fact, for portraiture I like my 24-70L better on crop than I do on FF. It works great for that purpose on my 7D's, but seems a little short on my 5DII. For portraits with the FF camera I prefer to use 85mm or 135mm primes or a 70-200mm zoom.
And I use a 300/4L a lot on crop cameras, too. That's a reasonable size and weight lens that allows me to shoot handheld and be pretty mobile. To frame a subject the same way with my FF camera I need to get out an 8 lb 500/4L instead, as well as a sturdy tripod to put it on. There's a big difference in price, too!
Some people are "L-coholics"... they won't even consider buying a lens that doesn't have a red stripe. Personally I think that's really silly. There are some excellent non-L and there are some L-series that simply don't impress me all that much.
For example, the macro 100/2.8 USM (non-L and non-IS version) is identical build quality to the 180/3.5L USM macro... Both have top image quality, are full 1:1 capable and internal focusing, have similar focus limiters... heck they even share the same tripod mounting ring (optional on the 100mm). Both have USM focus drive, too, though frankly it performs better on the 100mm than the 180mm (because the 100mm has f2.8 aperture). The 100/2.8 USM simply doesn't need fancy elements to render very high quality images, so it doesn't qualify to get a red stripe painted on it. The 100L IS macro is also an excellent lens, and having IS may make it a little more useful for handheld shooting. But IQ, most other performance factors, and build quality aren't all that better and I can't justify the 50% higher cost just for IS because I do most of my macro shooting with a tripod or at least a monopod, as well as often with a flash that freezes movement.
Other examples are the Tilt-Shift lenses. The 17mm and 24mm are L-series. The 45mm and 90mm are not. Yet they are all very equivalent in build and design. The two longer lenses just don't need fancy glass, so can't be L-series!
Finally, another example is the EF 24-105L IS USM versus the EF 28-135 IS USM. The L-series appears better built... but actually isn't all that much better in most respects. They have similar image quality (the 28-135 actually has less vignetting at the wide end, but isn't quite as sharp all the way out at 135mm). The 28-135mm is 1/3 stop faster at the wide end, but one stop slower racked out to it's longest focal lenght (is 2/3 slower at 105mm). Both are close focusing. Both have fast, accurate USM focus. Both have 3-stop IS that needs to be turned off when locked down on a tripod. Both tend to get "zoom creep". The 28-135 tends to have a little wink of play between the inner and outer barrel, but it doesn't seem to effect IQ at all. The L-series lens sometimes sees it's red stripe break and fall off (purely cosmetic). The 24-105mm sometimes has a ribbon cable break that causes AF and/or aperture control issues. I've has aperture issues with a 28-135, too, though. The 24-105L costs about 3X as much! I can buy the 28-135mm for $300 new (plus $25 for the hood, sold separately) or $200 used. The 24-105mm typically sells for $1000 new (hood included, like most L-series) or about $600 used.
I've got some L-series lenses... and some that aren't L-series. Whether or not it's an L is really one of the very last things I think about when choosing a lens. I buy a lens that will do all that I need it to do at the best price I can find. If it has a red stripe or a gold stripe or a silver stripe or doesn't even have any stripes at all, I could care less!
