I'll describe the typical Mode-S radar system based on my mostly European experience. Details will vary but the general concept applies world wide.
The radio signal received from the aircraft is digitized, CRC checked, compared to and combined with data from earlier transmissions from the aircraft. This data is usually encoded in ASTERIX format (typically category 048PDF) and transmitted over a datalink to a surveillance data processing system (SDPS) in the ATC centre. From the SDPS the data is further distributed to various client systems, among which the air traffic controller display system is one.
The radar processing itself is partly dedicated hardware and partly COTS (commercial-off-the-shelf) infrastructure. This is housed in a standard server cabinet. Network components are typically COTS high quality systems and duplicated for redundancy.
The datalink from the radar site to the ATC facilities can take many forms. Every country has its own regulator and local telecom market peculiarities, leading to a variety of implementations. Often a dedicated line will be used, usually with a backup line as well. Sometimes two telecom providers are used that have to provide fully independent data paths to the ATC centre. In a few cases radio links are used to connect surveillance systems, I know of surveillance systems (not radars) that use satellite links.
Within the ATC centre usually duplicated or triplicated networks are used for redundancy. These can be UTP or glassfibre networks. These are again based on COTS high availability/integrity components.