"Same size and shape" is a good first idea, but it is hard to check or prove. Geometry makes it precise with the rigid motions from Composing Transformations: two figures are congruent if some sequence of translations, reflections and rotations carries one exactly onto the other.
Why it works
Rigid motions keep every distance and every angle. So if a rigid motion carries figure $F$ onto figure $G$, each part of $F$ lands on a part of $G$ with the same length or measure. Congruent figures therefore have all corresponding parts equal. The definition also works for any shape, not just polygons: two circles, two curves, two letters of the alphabet.
Key idea
Two figures are congruent ($\cong$) when a sequence of rigid motions maps one onto the other. Then every pair of matching sides and angles is equal.
Worked example: Finding the motion
$\triangle ABC$ has $A(1, 2)$, $B(5, 2)$, $C(1, 4)$ and $\triangle DEF$ has $D(-2, -1)$, $E(-2, -5)$, $F(0, -1)$, with $A \to D$, $B \to E$, $C \to F$. Show the triangles are congruent.
- $\overline{AB}$ points right (4 units); $\overline{DE}$ points down (4 units). $\overline{AC}$ points up (2 units); $\overline{DF}$ points right (2 units). Right turns into down and up turns into right: that is a $90^\circ$ clockwise turn, $(x, y) \to (y, -x)$.
- Rotate: $A \to (2, -1)$, $B \to (2, -5)$, $C \to (4, -1)$.
- Each image is 4 units right of its target, so translate by $\langle -4, 0 \rangle$: $(2, -1) \to (-2, -1) = D$, $(2, -5) \to (-2, -5) = E$ and $(4, -1) \to (0, -1) = F$.
Answer: A $90^\circ$ clockwise rotation about the origin followed by a translation of $\langle -4, 0 \rangle$ maps $\triangle ABC$ onto $\triangle DEF$, so they are congruent.
Worked example: Showing two figures are not congruent
$\triangle PQR$ has sides $5$, $5$ and $6$. $\triangle STU$ has sides $5$, $6$ and $6$. Can they be congruent?
- A rigid motion keeps every length, so congruent triangles have the same three side lengths.
- The lists $\{5, 5, 6\}$ and $\{5, 6, 6\}$ differ.
Answer: No rigid motion can map one onto the other: they are not congruent.
Common mistake
Deciding congruence from one number, like equal areas or equal perimeters. A $2 \times 6$ rectangle and a $3 \times 4$ rectangle both have area 12, but no rigid motion maps one onto the other. Check matching lengths and angles, or find the motion.