System requirements are the translations of user requirements in a much more technical language. They are basically the things that a software must perform.
Not exactly. The system usually consists of hardware and software. In some situations, it could even include humans performing well defined processes (for instance, changing depleted batteries).
The requirements can be classified as User Requirements/System Requirements and Functional Requirements/Nonfunctional Requirements
In my understanding you are mixing classifications. There are at least user requirements; and system, hardware and software functional and non-functional requirements. You could even decompose the system into subsystems, or the software requirements in modules... Examples of the "high level analysis" requirements:
User requirement: The user should be able to turn off the alarm bell after acknowledging the error message.
System's non-functional requirement: The system should be able to attend up to 80 simultaneous requests by attaching a new processor.
System functional requirement: The system must alert the user with a 2500 Hz acoustic tone if the message contains the code 0xEFAC.
Hardware non-functional requirement: The system must tolerate a power outage for at least 10 minutes.
Hardware functional requirement: If the JP14 jumper is on and the register R4 has a 0xEFAC, the hardware must put +5V on PK1.
Software non-functional requirement: The software should be able to attend 50 simultaneous INFO requests.
Software functional requirement: The software must render a red alert box in the left corner if the message is corrupt.