National governments are responsible for raising taxes and setting tax rates. The amount of tax you pay is therefore decided by your national government, not the EU.
The EU's role is to oversee governments' national tax rules and decisions on tax rates (on company profits, personal income, savings and capital gains), to ensure that:
For some taxes, such as the value added tax (VAT) or taxes on petrol, tobacco and alcoholic beverages, all 27 national governments have agreed to set minimum tax rates so as to avoid a distortion of competition across borders within the EU.
No EU decisions on tax matters may be taken unless all member countries are in unanimous agreement.

Common rules on petrol tax ensure fair trade and prevent harmful tax competition
The EU also has nothing to say about how much countries spend - provided their budget balance and public debt stay within reasonable limits.
However, if countries overspend and go into too much debt, they could jeopardise economic growth in other EU countries and the stability of the eurozone.
The EU pays particular attention to company taxation because of the risk that tax breaks or very low tax rates in one country might unfairly lure firms away from competitor countries, or otherwise erode the tax base in other countries. EU countries are politically bound by a code of conduct not to do this.

Income taxes are decided by national governments.
For similar reasons minimum tax rates for VAT and excise duties on petrol, tobacco and alcohol have been agreed upon by all EU countries, and all are obliged to comply with the rules they have designed and agreed upon at the European level.
VAT is fundamental to a properly functioning single market and fair competition across the EU.
So the EU has set:
This still leaves considerable leeway for differences in national VAT rates: there are no upper limits on VAT rates. Although the general rule is that a single standard rate should be applied to all sales of goods and services, governments are also free to apply reduced rates to a narrow range of goods/services and some countries are temporarily exempted from some of the rules.
These products are subject to some common rules because differences in taxes (excise duties) can very easily distort competition across EU borders and trigger massive cross-border shopping at the expense of business in high-tax regions. Here too, the rules still leave plenty of room for variation, due to:
The EU has common rules on taxing energy products, so that it can take a unified approach to using taxes as incentives to energy efficiency. Again, the rules are flexible enough to accommodate special national circumstances.
Personal tax rules and rates are matters for your national government – except where your cross-border rights are affected. The EU rules are designed to ensure that you are not deterred from working in other EU countries by problems linked to the transfer and taxation of your pension and pension rights.
The EU also has a role in preventing cross-border tax evasion. EU governments lose legitimate revenue if their residents do not declare interest income on savings held abroad.
As an EU citizen, you can place your savings where you think they will get the best return, but you can't use this as a means of avoiding tax. Most European countries have agreed to share information on non-residents' savings.
The few exceptions (e.g. Austria, Luxembourg) instead impose a withholding tax, a large part of which they transfer to the saver's home country. As this is a bulk payment, the individual saver's anonymity is not breached, but the tax is still paid where it is due.
EU countries lose up to €1 trillion per year in revenue for their budgets because of tax fraud, tax evasion and the shadow economy. This corresponds to around 20% of all tax revenues. Although fighting fraud is each country's responsibility, coordinated action by all EU countries is needed as fraud is often organised across borders and anti-fraud measures taken in one country can have negative effects for Europe as a whole. The EU is currently examining an action plan to strengthen the fight against tax fraud and tax evasion.
The financial sector was a major cause of the economic crisis and received substantial government support over the past few years. A financial transaction tax can ensure that the sector makes a fair and substantial contribution to covering the cost of the crisis. Such a tax would also close the perceived "fairness gap" in taxation and public budgets. Eleven EU countries are now in the process of developing a common system for such a tax. This would allow them to go for a broad-based tax with significant revenue potential despite the high international mobility of financial transactions.