Tissue engineering can perhaps be best defined as the use of a combination of cells, engineering materials, and suitable biochemical factors to improve or replace biological functions.
While the semi-official definition of tissue engineering covers a broad range of applications, in practice the term has come to represent applications that repair or replace structural tissues (i.e., bone, cartilage, blood vessels, bladder, etc).
These are tissues that function by virtue of their mechanical properties.
A closely related (and older) field is cell transplantation.
This field is concerned with the transplantation of cells that perform a specific biochemical function (e.g., an artificial pancreas, or an artificial liver).
Tissue engineering solves problems by using living cells as engineering materials.
These could be artificial skin that includes living fibroblasts, cartilage repaired with living chondrocytes, or other types of cells used in other ways.