Enthalpy is a state function, meaning it is a property of a system, and not a process (like heat or work). It is defined as
$$H = U + PV,$$
where $H$ is the enthalpy, $U$ is the internal energy of the system, $P$ is the ambient pressure (assumed to be constant), and $V$ is volume.
Why is it defined in this way? Consider a gas in a closed box. The question is: how much energy does it take to create this system? The gas has internal energy $U$, which is the sum of the microscopic kinetic and potential energies of its constituent particles. But that's not the only energy it would have taken to create the system. We would have also had to do pressure-volume work to create the volume of the gas. Assuming the pressure $P$ is constant, and that we started from an empty volume ($V=0$), the work $W$ is equal to $PV$. So the minimum amount of energy necessary to create this box would be $U + W = U + PV$, which is what we call enthalpy
Why is it sometimes called the "heat content" of a system? Well, as with many quantities in thermo, we are often more interested in how it changes than its exact value. Consider the change in enthalpy
$$\Delta H = \Delta U + \Delta (PV) = \Delta U + P \Delta V$$
(since pressure is constant). The first law of thermodynamics states that, for a closed system, $\Delta U = W + Q$, where $Q$ is heat. The system does work $W = -P\Delta V$, which is negative since the system is doing work against its surroundings. Plugging these equations togethers, we get
$$\Delta H = W + Q + P \Delta V = -P \Delta V + Q + P \Delta V = Q.$$
So the change in enthalpy for a system is simply the heat added to the system (when all of our assumptions hold).