The 2038 bug is a problem with the calculation and storage of the dates in the hardware, in which time values are stored or calculated with a space too "limited" to contain a date later than 2038 (hence the name).
To be precise, the maximum date that can be represented is
03:14:07 UTC on January 19, 2038
The worst case of this bug is more or less the same as that which would have to hit the computer world for the "Millenium Bug" (Y2K).
The chances of catastrophic events are much greater, because the problem of representation is not bound to a human representation but to the computer's one. Power plants could shutdown for this reason, software on cars could not well calculate time derivates, or satellite communication systems that fail to restore a connection in case of connection loss. The Internet itself could be at risk, and some tablet/smartphone may not work.
Fortunately, there's plenty of time to mitigate these potential problems!
But you have to be aware of it.