Why do mosques have domes and minarets?

The first mosque had neither

The Prophet's ﷺ mosque in Madinah was a simple courtyard with walls of mud brick, pillars of palm trunks and a roof of palm leaves over part of it. Bilal gave the call to prayer from a nearby rooftop. What made it a mosque was not its shape but the people praying in it. The Quran puts the focus there too:

Why minarets

As cities grew, the call to prayer needed to reach more homes. Towers were built so the mu'adhdhin (caller) could climb up and be heard far away. Over time the minaret also became a landmark: you could see from across town where the mosque was. Today most adhans use loudspeakers, and in many Western cities the adhan isn't called outside at all, so minarets are mostly symbolic. Styles vary a lot: slim pencil towers in Turkey, square towers in Morocco and Spain, spiral ones in Iraq.

Why domes

Domes came from earlier building traditions in the region, including Roman and Byzantine architecture. Muslim builders adopted and refined them because a dome can cover a wide hall without many pillars, which keeps the prayer rows unbroken. A dome also helps the imam's voice carry across the hall and keeps the space cooler in hot climates. Its decoration, often geometric patterns and calligraphy, reminds worshippers of the vastness of the heavens without using images of people.

Not a religious requirement

Scholars agree that domes and minarets are permitted but not obligatory. Building a mosque at all is the virtue: the Prophet ﷺ said whoever builds a mosque for Allah, Allah builds for him a house in Paradise (Sahih al-Bukhari 450). He also warned against people showing off and competing over fancy mosques (Sunan Abi Dawud 449). So a modest, well-used prayer hall is better than a beautiful empty one.

In Europe and the USA, many mosques are converted houses, shops or warehouses with no dome at all, and some communities skip a minaret to fit in with local planning rules (Switzerland even banned new minarets by referendum in 2009). If your local mosque looks like an ordinary building, that's completely normal. See How do I find a convert-friendly mosque?