Both heat and work have intuitive definitions. However, we need to forego those as we study thermodynamics, because they can be misleading if not used carefully. To that end, we will rigorously define both concepts here.
Heat is the transfer of energy to a system via thermal contact with a reservoir. Work is the transfer of energy to a system via a change in the parameters of the system, such as volume.
This seemingly small distinction has significant consequences. Remember that a transfer of energy from a reservoir must obey the thermodynamic identity (taken for constant N and V), dU = τ dσ. Therefore a change in energy, i.e. a heat transfer, is accompanied by an entropy transfer. The addition of work, however, can't change the entropy of the system since we are only changing the external environment of the system.
We can look at the thermodynamic identity in a new way. The first term, τ dσ, can be thought of as the heat input, written dQ. The second term, - p dV, can be thought of as the work input, written dW. The third term, μ dN, can be thought of as the chemical work input, written dWc. Therefore the total change in energy is due entirely to the sum of the heat inputted, work, and chemical work done on the system.
We need to think of entropy in a new way, though it is yet the same fundamentally as before. Entropy cannot build up indefinitely in a system. If it is introduced accompanying some heat input, it must eventually be released from the system.
This restriction does not affect the conversion of work into work, however. A plant that converts the rush of a river into electricity does not have to worry about entropy. Similarly, conversion of work into heat does not lead to a buildup of entropy. Conversion of heat to work, however, the basic process of a heat engine, must be done carefully to avoid buildup of entropy.