What they are
Definition
Two fractions are equivalent when they stand for the same number.
Put another way: when they have the same irreducible fraction.
$\dfrac{3}{5}$
The unit is cut into $5$ pieces and we shade $3$ of them.
$\dfrac{6}{10}$
The same unit in $10$ pieces, shading $6$ of them. The shaded part is the same.
Both simplify to $\dfrac{3}{5}$, their common irreducible fraction.
The cross-product test
Test
Multiplying crosswise, the two fractions are equivalent if both products agree:
$$\dfrac{a}{b} = \dfrac{c}{d} \iff a \cdot d = b \cdot c$$
It is the quick way to check, with no need to simplify either fraction.
Example: finding the missing term
We look for the $x$ that makes the three fractions equivalent:
$$\dfrac{3}{5} = \dfrac{6}{10} = \dfrac{27}{x}$$
We apply the cross-product test to the last two:
$$10 \cdot 27 = 6 \cdot x \;\Longrightarrow\; 6x = 270 \;\Longrightarrow\; x = \dfrac{270}{6} = 45$$
Check: $\dfrac{27}{45}$, simplified by $9$, gives $\dfrac{3}{5}$.