Do Muslim men have to cover their heads?

Not required for men

A man's required covering is from the navel to the knee. The head is outside that, so there's no sin in going bareheaded at work, in the street or in the mosque. This is the position of all four schools (Hanafi, Maliki, Shafi'i and Hanbali). The head covering that is required in Islam is the women's hijab; for men it's a matter of custom and good manners.

What the Prophet ﷺ wore

The Prophet ﷺ usually covered his head, often with a turban. He entered Makkah on the day of its conquest wearing a black turban (Sahih Muslim 1358), and he sometimes wiped over his turban in wudu (Sahih al-Bukhari 205). Because of this, many scholars see covering the head as a recommended sunnah. Others say it was the custom of Arabs at the time, which is why they call it good but not a religious act in itself. Both views are respected.

In prayer

Allah tells us to dress well for prayer:

Many scholars, especially in the Hanafi school, say it's better for a man to cover his head in prayer and disliked to leave it bare out of laziness. Still, the prayer is valid either way. That's why many mosques keep a basket of caps at the entrance. See Can I pray in jeans?

Kufi, turban or nothing?

• Kufi or taqiyah: a small knitted or cloth cap. Easy, cheap and the most common choice in the West.
• Turban or shemagh: fine if you like it; some brothers wear one on Fridays. See What are the sunnahs of Friday?
• Baseball cap or beanie: allowed, though a normal cap's peak can get in the way of sujood, so turn it round or take it off.
• Bare head: completely fine; don't let anyone shame you.