Cells are the basic structural and physiological units of plants. Most plant reactions (cell division, photosynthesis, respiration, etc.) occur at the cellular level. Plant tissues (meristems, xylem, phloem, etc.) are large, organized groups of similar cells that work together to perform a specific function.
A unique feature of plant cells is that they are readily totipotent. In other words, almost all plant cells retain all of the genetic information (encoded in DNA) necessary to develop into a complete plant. This characteristic is the main reason that vegetative (asexual) reproduction works. For example, the cells of a small leaf cutting from an African violet have all of the genetic information necessary to generate a root system, stems, more leaves, and ultimately flowers.
Specialized groups of cells called meristems are a plant's growing points. Meristems are the site of rapid, almost continuous cell division. These cells either continue to divide or begin to differentiate into other tissues and organs. How they divide, and whether they ultimately become a tissue or an organ, are controlled by a complex array of internal plant hormones but also can be influenced by environmental conditions. In many cases, you can manipulate meristems to make a plant do something you want, such as change its growth pattern, flower, alter its branching habit, or produce vegetative growth.