(covers information from several alternate timelines)
An emergency team was a specialized group of Starfleet personnel formed to deal with emergency situations aboard Federation starships and space stations. Members from the medical staff were usually part (or all) of the team, which led to the often use of the terms medical emergency team, emergency medical team, or simply medical team. The team would usually traverse to the origin of the emergency and provide on-site assistance to try and stabilize the situation, and injured personnel may have required additional attention in a medical facility, such as sickbay. Occasionally, medical personnel were requested by simply asking for medical assistance instead of explicitly requesting an entire team, but the desired outcome was the same.
Captain Jean-Luc Picard called for a medical emergency team to transporter room 16 when Commander William T. Riker, Lieutenant Commander Data and Lieutenant Worf were beamed back to the USS Enterprise-D. (TNG: "We'll Always Have Paris")
In an alternate timeline 2366, Riker told Captain Rachel Garrett of the USS Enterprise-C that emergency teams were boarding the vessel to assess the damage done to it after the battle with Romulan warbirds at Narendra III. (TNG: "Yesterday's Enterprise")
As part of the release of hostages in Ten Forward being held by the aliens inhabiting members of his crew, Picard called for an emergency medical team from Sickbay to accompany him to Ten Forward. He later called for a team to the cargo bay when the aliens were transported away. (TNG: "Power Play")
The Emergency Medical Holographic program was one of Starfleet's initiatives to provide medical assistance in dire circumstances, but was normally limited to only areas that supported holographic projections, such as sickbays aboard Intrepid-class starships. (Star Trek: Voyager)