There's one nutrient that all vegans should be aware of--vitamin B12!
B12 is not only beneficial (it can help with memory loss, diabetes, depression, osteoporosis, IBS, and many diseases) it is also an essential vitamin, that your body needs to function. It's important for your brain, blood, skin, hair and nerve cells, among other things. It is even said to help protect against cancer and heart disease.
Where can you get some, you might ask? That's where things get a little tricky for people eating primarily plant-based diets--B12 comes from animal products, including meat, milk, and eggs.
If you eat a plant-based diet, you'll likely need to rely on fortified food products (such as many soy products and cereals) and possibly a B12 supplement. Finding out if vitamin B12 has been added to a food product is simple enough, you can just check the nutrition fact panel on the food label.
One of my favorite sources of B12 is nutritional yeast (pictured at the top of the page, and used in the scramble below). This is a staple in many vegan cheeze recipes because of it's naturally cheesy flavor. Most recipes however, call for relatively small amounts. Getting your recommended daily value from nutritional yeast alone can be tough, and the amount of B12 can vary from brand to brand, so you'll definitely want to check the label to see how much B12 is in a serving.
Mushrooms can contain B12, as do fermented foods such as tempeh or miso, but the amounts can be quite low. Indeed, many agree that it is very difficult for vegans to get the required amount of B12 from plant-based foods alone. Additionally, Vitamin B12 is better absorbed by the body when it is taken along with other B vitamins, such as niacin, riboflavin, vitamin B6, and magnesium. Multivitamins and supplements are therefore a great option for vegans--and most vegan supplements will include a healthy dose of B12.
If you do not wish to take a supplement or do not think you need it, you can have your levels checked at your doctor's office. Be sure to ask about how often you should have them checked. Your body can store B12 for years, so if you have eaten animal products in the past, it is possible that your body is still storing B12, but these levels can change over time.
For more information, see this article from the Vegan Society and this one about Vegan Sources of Vitamin B12, which lists specific foods that are fortified with or contain B12.