Tax class has an enormous impact on your monthly net income. With a gross salary of €4,000, the difference between tax class I and tax class III can be more than €400 per month. That's almost €5,000 per year, available to you either immediately or only after filing your tax return.
Which tax class is optimal for you? If you're single, you automatically have tax class I – nothing can be changed there. It gets interesting for married couples: the combination III/V is worthwhile if one partner earns significantly more (e.g. €5,000 vs. €2,000). The higher earner takes III, the other V. With roughly equal salaries, IV/IV is better. Careful: the tax class only changes your monthly advance payment, not your annual tax liability!
Especially important before parental leave: parental allowance is based on your net income. If you're planning to go on parental leave, switching to tax class III (for the partner going on leave) can bring in several thousand euros more in parental allowance. The switch needs to happen in time though – at least 7 months before the birth is ideal.