I was gifted the plant about 3 years ago and don't know what it is. It's grown a ton in that time, but no sign of fruit or flowering at all. This is the first plant I've been able to keep alive and I'm very attached to her! I want to be sure I'm taking care of her perfectly. :) (Reverse image search on Google says Prickly Pear, but I highly doubt that's it.) Any help is appreciated]1
It's definitely an Opuntia. I think it's a south American variety. Possibly O. monacantha. You don't mention anything about the conditions this plant enjoys, but based on the picture I'd guess it lives indoors and probably has never been fertilized.
In order to flower and grow vigorously it will need plenty of sun and some fertilizer. You can use a balanced fertilizer at half strength a couple of times a year.
Opuntias generally flower in the spring. They put out new pads and flower buds at the same time.
Most Opuntia are not especially sensitive to watering. I would recommend watering it when the top inch or so of soil is dry (dryer if you are using regular potting soil instead of cactus-specific soil). In summer, water deeply, in winter, less so. Keep in mind that it will dry out much more slowly in winter.
Don't be shy about pruning pads to keep the plant compact, if that's your thing. If you let it grow without pruning, eventually it parts will begin to droop and/or snap off. It's what they do.
One final note, the hair (glochids) are extremely irritating; some people are allergic. If you get some on you the best method to remove them is to put some white glue (like Elmer's) on the spot, let it dry and peel it off.
I think Google is right. There are many species of opuntia (prickly pear) with a wide range of different shapes and sizes and your plant looks like one of them IMO. I don't know which species, though.
Some species can grow to tree-sized plants, and don't flower until thy are 10 or 20 years old, but to encourage yours to flower, keep it in full sunshine if possible. For the summer 7 or 8 months, water it once or twice a week to keep it growing fast. Then stop watering completely for 4 or 5 months over winter.
The plant will "shrink" as it uses up its stored water supply, but don't take pity on it and "water it a little bit in case it dies." The whole idea is to make it go dormant. A plant as big as yours would probably survive 12 months in full sunshine with no water at all, though it wouldn't look very pretty at the end of that time.
If you get lucky, near the end of the "dry spell" it will start to develop some flower buds as its "last chance" to produce seed in case it really does die from lack of water. If that happens, start watering it again. Don't overdo the watering for the first few weeks, otherwise the plant will take up a lot or water quickly and some of the pads may split open. After you start watering it will soon start growing fast again.
The best time to repot it is at the end of the "dry period". Repotting opuntias can be painful, because even if the spines and hairs are short (as on your plant) they are stiff enough to penetrate the skin and break off, and they can cause a lot of irritation until you get rid of all the broken ends. Wear thick gardening or "industrial" gloves. Household rubber gloves that you wear for cleaning etc, give no protection at all!