Phones use an integrated circuit called a System On a Chip (SOC). As the name suggests this integrates most of the components needed by the phone into one component. Some of the common SOCs used in phones are Qualcomm (e.g. Snapdragon 835), MediaTek (e.g. Helio X30) and Samsung Exynos.
A SOC includes the CPU cores (main processor), GPU and video (graphics), cellular modem, memory controller, display support, camera support, multimedia, and connectivity.
CPU cores are mainly provided by ARM (e.g. Cortex A53), Qualcomm (e.g. Kryo 280) and Samsung. The higher the number of cores and the higher their frequency, generally speaking, the faster the phone is (more operations can be performed). The CPU cores are typically arranged in clusters (groups). This allows better power management (e.g. 8 CPU cores may be arranged as 2 x 4-core groups, where 1 group may have a lower clock frequency to reduce power usage and temperature). The speed (base frequency multiplier) of each CPU core can be reduced or increased based on the workload, power, and temperature. Phone spec. sheets typically show the maximum CPU core frequency.
It is interesting to note that some phones very quickly hit their thermal limit (e.g. 100 °C in 5 seconds) and then throttle (slow) the CPU frequency to a lower value to reduce power and hence reduce the temperature. This can be impacted by running apps, the ambient temperature, and the case.
The GPU (graphics) core is typically provided by ARM (e.g. Mali G71) or Qualcomm (e.g. Adreno 540).