This is the question I get asked most often, nearly everybody asks when I tell them I'm a vegan! The answer is: from plants! Nuts, lentils, peas, avocado, all different types of beans (like baked beans, butter beans, cannellini beans, pinto beans, black beans, refried beans etc.), almond butter, tofu, peanut butter, edamame, soya milk, quinoa, tempeh and faux 'meat' products are all high in protein. You can get beans and pulses in tins or microwaveable pouches that can be used straight away with no soaking required, or you can even sprout dried beans and pulses like my friend Keith (who isn't even a vegan!) It's very easy to cook using these ingredients so I make lots of traditional meals but substitute the vegan equivalent for the meat.
It's really not a big deal to get enough protein, and protein requirements are actually a lot lower than most people think. Recommended amounts have been halved in the last 20 years and more and more diseases are being linked with eating too much animal protein. The WHO (World Health Organisation) recommends no more than 8% of our energy (calories) should be from protein.
It's also worth remembering that there are successful vegan body builders, runners, Olympic athletes, cyclists etc. (like Carl Lewis, Venus & Serena Williams, Mike Tyson, Molly Cameron, Fiona Oakes, Maria Navratilova and many more) who are obviously getting enough protein and are at optimal health!
A few people have misguidedly informed me that meat, dairy and eggs are 'higher quality' protein, which is simply untrue; you can get all the essential amino acids from plant proteins. I have also had a few people tell me that the only way to get enough protein on a vegetarian or vegan diet is to combine different proteins which is off-puttingly complicated, but this opinion is actually based on out of date research which has been disproved, and even the author of the book has since repudiated her claims.
In it's Journal of the American Dietetic Association, the American Dietetic Association, one of the world's leading health bodies, sets out its position that "well-planned vegetarian diets, including total vegetarian or vegan diets, are healthful, nutritionally adequate, and may provide health benefits in the prevention and treatment of certain diseases. Well-planned vegetarian diets are appropriate for individuals during all stages of the life cycle, including pregnancy, lactation, infancy, childhood, and adolescence, and for athletes. [...] This article reviews the current data related to key nutrients for vegetarians including protein, n-3 fatty acids, iron, zinc, iodine, calcium, and vitamins D and B-12. A vegetarian diet can meet current recommendations for all of these nutrients",
For more in-depth, referenced information I highly recommend the Protein Myth factsheet and reading The China Study.