A thermodynamic process involves the change in any of the sate variables of a thermodynamic system such as pressure $p$, temperature $T$ or volume $V$. In a broad sense of thermodynamic processes we consider reversible and irreversible processes and we distinguish various other thermodynamic processes based on their special properties.
Reversible and Irreversible Process
When one state of a thermodynamic process changed into another state which can be reversed to the initial state along the same path is reversible thermodynamic process.
A reversible thermodynamic process takes place in an infinitesimal change in the state which can be reversed back by the same but opposite infinitesimal change. For example, you added infinitesimal heat $dQ$ into a system and the system undergoes an infinitesimal change which can be reversed back by removing the same heat $dQ$ from the system.
A reversible process takes place in thermodynamic equilibrium . Due to the infinitesimal change in the state of the system, we say that reversible process takes place in thermodynamic equilibrium and therefore it is also called quasi-static process. A reversible process in thermodynamic equilibrium is an idealization but if the change in sate is made extremely small, the process is nearly in thermodynamic equilibrium and can be reversed by the same opposite change.
On the other hand there are irreversible processes which can not be reversed and therefore they are called irreversible processes. If a thermodynamic process is irreversible we don't know the actual path of that process or not easily predictable and can not be reversed along the same path in its initial state. Even if you bring an irreversible thermodynamic process to its initial state, the paths will not be the same and the initial conditions of the process can not be precisely attained. For example the processes such as free expansion of gas in a cylinder, work done by friction are irreversible processes.
Kinds Thermodynamic Process
Every thermodynamic process can be either reversible or irreversible. We define other kinds of thermodynamic processes based on their special properties which are adiabatic, isothermal, isovolumetric and isobaric processes.
In adiabatic thermodynamic process no heat can enter or leave the system, that is $Q = 0$. When $Q = 0$, the first law of thermodynamics tells us that $W = -\Delta U$, that is, the work done by the system in adiabatic process is equal to the decrease in internal energy of the system. Again if work is done on a system adiabatically the internal energy increases as suggested by $\Delta U = -W$. Note that negative work means the work is done on the system.
You already know that if a system contains only the ideal gas, the internal energy of the system depends only on the temperature of the system and work done by the system in adiabatic process decreases the the internal energy and hence decreases the temperature of the system. And if work is done on the system, the internal energy increases and the temperature increases.
The thermodynamic process in which the temperature remains constant is called isothermal process. According to the first law of thermodynamics we have $Q = \Delta U + W$. The heat added in a system in isothermal process is used to increase the internal energy and do work on the surroundings but isothermal process is a slower process in order to keep thermal equilibrium (constant temperature). It means the process should be slower enough to make the temperature constant during the process.
In case of an ideal gas the internal energy depends only on the temperature of the system and there is no change in internal energy ($\Delta U = 0$). So, all amount of heat added in the system is used to do work by the system containing provided that the number of moles of the ideal gas is constant. We consider that the number of moles of a gas always remains constant in a particular process and we don't talk about it later; you should think it a constant quantity in further discussions. As the temperature $T$ remains constant, the ideal gas equation $pV = nRT$ shows that the quantity $pV$ is constant. Let the pressure and volume at the initial state be $p_1$ and $v_1$ and at the final state be $p_2$ and $v_2$ respectively. And you can show that
\[p_1V_1 = p_2V_2\]
In isovolumetric process the volume of the process remains constant. And if the volume is constant, the work done $W$ is zero. Therefore, according to the first law of thermodynamics, in isovolumetric process $Q = \Delta U$. So, the heat added in the system is used to increase the internal energy of the system. Isovolumetric process is also called isochoric process.
In isobaric process the pressure of the system remains constant. As the pressure remains constant the work done is $W = p(V_f - V_i)$ where $V_i$ and $V_f$ are initial and final volumes respectively. As only the pressure remains constant, the first law of thermodynamics is as usual which is $Q = \Delta U + W$.