tldr; The Czech president is not a figurehead.
He has real political power. Without approval from the government, he can appoint the heads of some agencies. He can dissolve parliament. With the approval of the government, he can negotiate international treaties and command the armed forces. He can speak at will in parliament and has the right to attend cabinet meetings. Subject to approval by the Senate, he can appoint Supreme Court judges.
What is not that uncommon, compared to other EU countries is his right to appoint the prime minister. It is checked by the obligatory vote of confidence the government has to pass in parliament - a procedure for example matched in Italy.
Since everyone answering so far has only referred to secondary sources, let me quote the Czech constitution itself. Wikisource helpfully has an English translation. These are the responsibilities of the president:
Article 62 lists what the president can do without further approval from the government:
President of the Republic shall
a) appoint and recall the Prime Minister and other members of the Government and accept their resignation, recall the Government and accept its resignation;
b) convene sessions of the Chamber of Deputies;
c) dissolve the Chamber of Deputies;
d) authorize the Government the resignation of which the President has accepted or which he has recalled to execute their office temporarily until a new Government is appointed;
e) appoint Justices of the Constitutional Court, its Chief Justice and Assistant Chief Justices;
f) appoint from among the Justices of the Supreme Court the Chief Justice and Assistant Chief Justices of the Supreme Court;
g) pardon and mitigate penalties imposed by the court and expunge sentences;
h) have the right to return to the Parliament an enacted law with the exception of Constitutional Acts;
i) sign enacted laws;
j) appoint the President and the Vice-President of the Supreme Control Office;
k) appoint members of the Bank Board of the Czech National Bank.
Article 63 lists what the president can do if countersigned by the government:
(1) President of the Republic shall furthermore
a) represent the State with respect to other countries;
b) negotiate and ratify international treaties; he may delegate the negotiation of international treaties to the Government or, subject to the Government consent, to its individual members;
c) be the Commander in Chief of the Armed Forces;
d) receive heads of diplomatic missions;
e) appoint and recalls heads of diplomatic missions;
f) call elections to the Chamber of Deputies and the Senate;
g) appoint and promote generals;
h) confer and award state decorations, unless he authorizes other body to do so;
i) appoint judges;
j) order not to initiate criminal proceedings and suspend them if they are already initiated; and
k) have the right to grant amnesty.
(2) President of the Republic shall be also entitled to exercise powers not explicitly defined in the Constitutional Act, if the law provides so.
(3) Decisions made by the President of the Republic pursuant to the provisions of paragraphs 1 and 2 shall be valid only if countersigned by the Prime Minister or by a member of the Government so authorized by the Prime Minister.
(4) Responsibility for a decision made by the President of the Republic, which must be countersigned by the Prime Minister or a member of the Government authorized by him, shall be borne by the Government.
Article 64 gives the president the right to take part in discussions of the government and the parliament:
(1) President of the Republic has the right to attend meetings of both the Chambers of the Parliament, their Committees and Commissions. He shall be given the floor whenever he asks for it.
(2) President of the Republic has the right to attend meetings of the Government, to ask for reports from the Government and its members, and to discuss with the Government or its members the issues that are in their competence.
Article 68 details the way the memebers of government are choosen:
(1) The Government shall be accountable to the Chamber of Deputies.
(2) The Prime Minister shall be appointed by the President of the Republic who shall appoint on the Prime Minister's proposal the other members of the Government and shall entrust them with the direction of individual ministries or other agencies.
(3) Within thirty days after its appointment the Government shall present itself to the Chamber of Deputies and shall ask it for a vote of confidence.
(4) If a newly appointed Government fails to win the confidence of the Chamber of Deputies, the procedure specified in paragraphs 2 and 3 shall be followed. If a thus appointed Government again fails to win the confidence of the Chamber of Deputies, the President of the Republic shall appoint a Prime Minister on the proposal of the Chairman of the Chamber of Deputies.
(5) In other cases the President of the Republic shall appoint and recall on the proposal of the Prime Minister the other members of the Government and shall entrust them with the direction of ministries or other agencies.
There are some more provisions following for cases where the government does not win a vote of confidence, the prime minister wants to sack a minster, a minister resigns or comparable cases. They all go through the President, but either he has to follow the proposal of the prime minister, or he is legally required to act one way or the other.
Article 84 describes the appointment of judges for the constitutional court:
(2) Justices of the Constitutional Court shall be appointed by the President of the Republic and shall be confirmed by the Senate.