Algebraic fractions follow the same rules as numerical fractions but with expressions instead of numbers.
Simplifying: factorise the numerator and denominator, then cancel common factors. $$\frac{x^2 - 4}{x^2 + 2x} = \frac{(x-2)(x+2)}{x(x+2)} = \frac{x-2}{x}$$
Adding/subtracting: find a common denominator, adjust numerators, combine. $$\frac{3}{x} + \frac{2}{x+1} = \frac{3(x+1) + 2x}{x(x+1)} = \frac{5x+3}{x(x+1)}$$
Multiplying: multiply numerators and denominators, then simplify. $$\frac{2x}{3} \times \frac{6}{x^2} = \frac{12x}{3x^2} = \frac{4}{x}$$
Dividing: keep, change, flip. $$\frac{x^2}{4} \div \frac{x}{2} = \frac{x^2}{4} \times \frac{2}{x} = \frac{2x^2}{4x} = \frac{x}{2}$$
Common error: cancelling terms that are being added (not multiplied) — you can only cancel factors that appear in the whole numerator and the whole denominator.