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As there are many types of satellites and their purposes, there are many antennas used to receive telemetry data from satellites, to communicate with satellites, etc.
Take low-orbit satellites. They are mostly used for scientific experiments above the atmosphere. Such satellite instruments are either programmed to continuously transmit data to Earth-located receivers, or, to receive commands as needed. The distance from Earth to a satellite is several thousands miles and data stream is "slow". Usually, UHF or VHF is used for transmission, and Yagi antennas (sometimes crossed) are used.
Medium-orbit satellites like GPS and data-distribution types can use helical antennas as well as their corresponding opposites on Earth. GPS antennas are often miniaturized and often hidden inside of receiver plastic case.
Geo-synchronous satellites are used for global or regional wideband communication. The power budget in uplink and downlink is often power-short, mainly when broadband channels are used. High-gain antennas are needed, and microwave bands like 4/6 or 12/14 GHz are used. At microwaves, parabolic antennas are the best; large ones for base stations like Intelsat, small dishes for receiving customers.
As there are now many different systems in use, find good textbooks on different types of antennas and systems. To my knowledge, no book covers all.
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