JT has the answer.
Stills aren't used very much(to my knowledge anyway) in the FDNY, but in many some other departments, a Still Alarm can be either one or two companies notifying fire dispatch that they are responding to a call w/o being dispatched to it(like a verbal alarm or "walk-in" in the FDNY). Or, it is a minor call in which one or two companies respond to, such as a small fire or other minor incident.
Here's the break-down of a Still Alarm in Chicago:
In the Chicago Fire Department, a Still Alarm refers to the general response to a reported structure fire. A "Still" as it's called in Chicago gets 2 & 2 w/a Battalion Chief. If it's a job, the Still gets a Squad(CFD's version of Rescues), a Command Van, and a R.I.T. Response(1 Truck for R.I.T. and 1 Battalion Chief for the R.I.T. Chief, as well as an ALS Ambulance and an EMS Field Officer), thus, a Still Alarm job would get 2 Engines, 3 Trucks(1 for R.I.T.), 2 Battalion Chiefs(1 for R.I.T.), 1 Squad, and 1 Command Van. If the Squad is nearby the fire, it goes on the initial Still w/the 2 & 2 and the Chief.
A Box Alarm is different. It gets 4 & 2 w/a Chief and is not an upgrade from a Still. It is the assignment for a pull box activation near a high-risk area(i.e., nursing homes, hospitals, government buildings, etc.). It can be upgraded to a Still and Box if it actually turns out to be a fire.
There is also the Still and Box Assignment, which is requested by an officer on scene or can be dispatched by the Fire Alarm Offices on certain occasions. It is basically an upgrade of the Still, but technically, not a Working Fire. To my knowledge, if the Still is upgraded to a Still and Box, 2 additional Engines, a Tower Ladder, 2 additional Chiefs, a Deputy District Chief, a Squad, an Ambulance, a Command Vann, a OFI Unit(Fire Marshal), and an EMS Field Officer, thus, a Still & Box Assignment, w/o a R.I.T. Response gets 4 & 2, a Tower Ladder, 3 Battalion Chiefs, a Deputy District Chief, a Squad, an Ambulance, a Command Van, an OFI Unit, and an EMS Field Officer.
Finally, there is the High-Rise Still Assignment, the structure fire assignment for fires in high-rise buildings(obviously). It gets 4 & 4, 3 Battalion Chiefs, a Squad, an ALS Ambulance, and an EMS Field Officer. If not sure if it can be upgraded to a Still and Box, though, but it is not an upgrade of a normal Still.
