BlackBerry is a common noun, meaning either a 
small berry in the 
rubus genus, or more commonly today, any 
mobile phone that has support for 
real-time or 
semi-real-time push email.
The first company to coin the term 
BlackBerry was 
Canadian-based 
Research In Motion, but by no means are they the only company to produce BlackBerries which has through aggressive use as a noun by both RIM and its partners (noticeably 
Verizon Wireless who uses the tag 
Sent from my Verizon Wireless BlackBerry and uses the term as a noun in its advertisements).
Other companies that currently make BlackBerries include 
Nokia, 
HTC, 
Sony-Ericsson, 
Motorola and a host of other mobile phone companies.  It's argued that to be classified as a BlackBerry, you also need to have a slow, unstable, and basic web browser, but this appears to no longer be the case as even the 
Research in Motion mobile devices have announced that they will be switching to a 
webkit-based browser making it the last popular maker of BlackBerries to do so.