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Triangles are congruent if their corresponding sides and angles are equal. It is necessary to understand that these triangles can go through isometry( translations, rotations and reflections), and still be the same, but cam NEVER change size or have different angles and still be considered congruent. In Layman's Terms, two triangles are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply moved).
Usually it is sufficient to establish the equality of three corresponding parts and use one of the following results to conclude the congruence of the two triangles(1):
Side Angle Side Method: Two triangles are congruent if a pair of corresponding sides and the angle between are equal.(1)
Angle Angle Side- Two triangles are congruent if the two angles and one corresponding angle are equal(this is similar to the Side Angle Side method, but uses a different acute angle).
Angle Side Angle Method: Two triangles are congruent if a pair of corresponding angles and the included side are equal. (1)
The Ambiguous Case(AKA Side Side Angle method): This method does not guarantee congruence, but gives two triangles, of which one is congruent and the other is not.\
This method is only valid in two conditions: 1) The Hypotenuse-Leg condition. This is true because all right triangles (which this condition is used with) have a congruent angle (the right angle). If the hypotenuse and a leg opposite to the hypotenuse of that triangle are congruent to the hypotenuse and leg of a different triangle, the two triangles are congruent.
2) If the angle and sides which are known to be equal, the side opposite the angle is longer than the other side, SSA is also valid