NOTE: this answer only focuses on proteins.
With beans (beans, chickpeas, lentils, etc.) and cereals.
There are proteins in many vegetable foods, the difference with meat is the percentage of protein. Here you have a table of protein content of most protein-rich plant foods, plus beef meat for comparison (values are g of protein per 100 g of uncooked product):
- beef meat 22
- beans 24
- chickpeas 21
- lentils 23
- wheat (bread, pasta, etc.) 10-13
- corn 10
- rice 6-8
In a omnivore diet, meat is just one source of proteins among others, and provides just a part of total protein intake. It's perfectly possible to get enough proteins without meat. Actually our society is suffering from diseases that are caused by excess, not lack, of proteins [3,4].
Essential amino acids
Every protein we eat through the diet is made of amino acids. There are some amino acids we need to eat in order to satisfy our requirements, these are called essential amino acids.
While meat contains all essential amino acids, common vegetable foods -individually- lack some essential amino acid. The essential amino acids that are not found in some vegetable foods can be found in other vegetable foods, for this reason it's important to eat variedly. If focusing on one type of bean, like soy, can lead to deficiencies, we avoid them by rotating the consumption of beans and cereals over the week. Some examples can be found on the web.
What was said until now is also valid for vegan diets: diets that exclude any animal product as meat, fish, milk, dairy, eggs, honey.
With diets based on vegetable foods it's perfectly possible to satisfy all nutritional needs 'during all stages of the life cycle, including pregnancy, lactation, infancy, childhood, and adolescence, and for athletes', as stated by the American Dietetic Association in 2009. You just have to eat enough beans, cereals and vegetables (they contain proteins too).
 Report of a Joint WHO / FAO Expert Consultation. (2003). Diet, nutrition and the prevention of chronic diseases. World Health Organization technical report series (Vol. 916). doi:ISBN 92 4 120916 X ISSN 0512-3054 (NLM classification: QU 145)
 World Cancer Research Fund, & American Institute for Cancer Research. (2007). Food, Nutrition, Physical Activity, and the Prevention of Cancer: a Global Perspective. Cancer Research.