The governmental body which is responsible for the registration of trademarks and other intellectual property is the Intellectual Property Office which is usually a branch of the Department of Commerce of the given country.
Additionally, other government bodies can be found in the US if you are looking for trademark registration in a specific state. In the EU, you can register with a national IPO (for country-wide protection) or with the European Union Intellectual Property Office (for protection at the EU level).
Please note that with the rising number of trademark applications, scam attempts have also increased. These take the form of posing as a government body (with a name similar to that of the IPO you are registering with) and sending invoices for additional payments. We recommend always checking the correct name and abbreviation of the office handling your registration and disregarding any other communication.