Enthalpy is a state function. This means that its value is only dependent upon the current state and properties of the system and is completely independent of how the system arrived at this state. Since enthalpy is a state function, this reinforces Hess's Law:

The standard enthalpy change for any process is the same, no matter which route the process took. The system has a specific enthalpy value at A and another different value at B and again a different value at C.

Enthalpy is defined in terms of the internal energy, pressure and volume of the system:

H = U + pV

If the system is maintained at constant pressure throughout a process, then the heat supplied to the system is equal to the enthalpy change of the system. (This is the case when the system is free to expand and does no other work.)

We can prove this from H = U + pV: 

H = U + pV
\H + dH = U + dU + (p +dp)(V - dV)

since at constant pressure dp = 0
\H + dH = U + dU + pV - pdV

since H = U + pV
\dH = dU - pdV

since dU = q + w
\dH = q + w - pdV

since w = pdV
\dH = q

Since dH = dU + pdV, changes involving solids and liquids, where dV is negligible, dH ≈ dU. However, changes involving perfect gases, we can use the equation of state, pV = nRT, giving
H = U + nRT, so if a change involves a change in the number of moles of gas, we can relate this to the difference between the enthalpy change and the internal energy change.

The fact that dH = q implies that if we supply heat to the system, the enthalpy of the system will increase. This is indeed true, and we have that