Your organization most likely already has systems generating events, such as syslog messages and network traps. Event management is the idea that the events from all these systems could be put together, correlated, and then incidents or other records generated as appropriate.
Event management tracks automatically-generated events in your environment that are "significant for the management of the IT Infrastructure" or otherwise relevant for services being delivered. It's up to your institution to decide what's "significant." Event management then filters and classifies the events into one of three categories:
Ideally event management performs correlation--the buzz-word here is a "correlation engine." This correlation should help people understand that event #2 (service is back up) is related to to event #1 (service went down). This correlation could also work between systems, e.g. a server's database and applications are generating exceptions because the underlying server is generating exceptions.
Event management can call incident management: exceptions and warnings could then generate incidents for real people to review.
Conversely, other processes can query event management: incident management and problem management could also search the event history for more information about why something broke.
Event management is ideally executed by an automated system. People may design and query the event management tool, but no person reviews every event that is generated.
Event management cannot stand on its own--it needs to work closely together with Service Design to ensure systems are designed to generate appropriate event, and it needs to work with Incident Management and perhaps even Change Management to ensure that appropriate events are escalated.
Vendors want to sell event management tools. These tools need to fit well within your ITSM suite.
Make sure that any event management tool you support can receive events from the systems you already use. An event management tool should be able to gather events from your other systems tracking events.