All living things contain DNA (well most, maybe not a prion, are prions living?). The function of DNA is to manufacture protein, that's it. So, all living things contain protein. Those proteins are found in structural components (muscles, plant cell walls), in chemicals like hormones, enzymes, immune cells, all over the place. Plant foods, as well as animal foods, provide these proteins to the human diet.
There are some differences between these food sources.
One, protein consists of amino acids. And each living thing we eat provides those amino acids in different ratios. (Which is why it's important to eat a variety of foods, although, as I showed here, you can get all your essential amino acids if all you ate were potatoes.)
Two, protein in vegetable matter is usually less bioavailable because of other compounds it comes packaged with. Where we might absorb 95% of the protein from animal foods, we might absorb 90% from plant foods. Lots of factors affect absorption though, so these are round figures.
The amount of protein we need is still debated. It's generally accepted that humans need about 0.8g/kg/day - that's 44g protein for a 120 lb person, 65g for a 180 lb person. And, of course, it varies depending on age, gender, life stage, body composition, etc.