If a system loses 250 kJ of heat at the same time that it is doing 500 kJ of work, what is the change in the internal energy of the system?

It refers to energy contained within the system, while excluding the kinetic energy of motion of the system as a whole and the potential energy of the system as a whole due to external force fields. It keeps account of the gains and losses of energy of the system.
For a system undergoing only thermodynamics processes, i.e. a closed system that can exchange only heat and work, the change in the internal energy is

d U = delta Q + delta W

which constitutes thefirst law of thermodynamics
therfore according to the question
du= (-250-500 )kj
du=-750kJ
as the system loses heat thus it is negative and as work is done by it thus it is also negative