Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.
In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure. The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc.
In the diagram above, the triangles are drawn next to each other and it is obvious they are identical. However, one triangle may be rotated, flipped over (reflected), or the two triangles may share a common side. These cases are discussed further on other pages:
One way to think about triangle congruence is to imagine they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in various ways so they make a pile where they exactly fit over each other.
If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see Why AAA doesn't work.
They are called similar triangles (See Similar Triangles).
Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence. See Why SSA doesn't work.
If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.
To remember this important idea, some find it helpful to use the acronym CPCTC, which stands for "Corresponding Parts of Congruent Triangles are Congruent".
In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.