First, let me introduce some terminology. A critical section (CS) is a sequence of instructions that can be executed by at most one process at the same time. When using critical sections, the code can be broken down into the following sections:
// Some arbitrary code (such as initialization).
EnterCriticalSection(cs);
// The code that constitutes the CS.
// Only one process can be executing this code at the same time.
LeaveCriticalSection(cs);
// Some arbitrary code. This is called the remainder section.
The first section contains some code such as initialization code. We don't have a name for that section. The second section is the code that tries to enter the CS. The third section is the CS itself. The fourth section is the code that leaves the critical section. The fifth and last section is called the remainder section which can contain any code. Note that the CS itself can be different between processes (consider for example a process that that receives requests from a client and insert them in a queue and another process that processes these requests).
To make sure that an implementation of critical sections works properly, there are three conditions that must be satisfied. You mentioned two of them (which I will explain next). The third is mutual exclusion which is obviously vital. It's worth noting that mutual exclusion applies only to the CS and the leave section. However, the other three sections are not exclusive.
The first condition is progress. The purpose of this condition is to make sure that either some process is currently in the CS and doing some work or, if there was at least one process that wants to enter the CS, it will and then do some work. In both cases, some work is getting done and therefore all processes are making progress overall.
Progress: If no process is executing in its critical section and
some processes wish to enter their critical sections, then only those
processes that are not executing in their remainder section can
participate in deciding which will enter its critical section next,
and this selection cannot be postponed indefinitely.
Let's understand this definition sentence by sentence.
If no process is executing in its critical section
If there is a process executing in its critical section (even though not stated explicitly, this includes the leave section as well), then this means that some work is getting done. So we are making progress. Otherwise, if this was not the case...
and some processes wish to enter their critical sections
If no process wants to enter their critical sections, then there is no more work to do. Otherwise, if there is at least one process that wishes to enter its critical section...
then only those processes that are not executing in their remainder section
This means we are talking about those processes that are executing in either of the first two sections (remember, no process is executing in its critical section or the leave section)...
can participate in deciding which will enter its critical section next,
Since there is at least one process that wishes to enter its CS, somehow we must choose one of them to enter its CS. But who's going to make this decision? Those process who already requested permission to enter their critical sections have the right to participate in making this decision. In addition, those processes that may wish to enter their CSs but have not yet requested the permission to do so (this means that they are in executing in the first section) also have the right to participate in making this decision.
and this selection cannot be postponed indefinitely.
This states that it will take a limited amount of time to select a process to enter its CS. In particular, no deadlock or livelock will occur. So after this limited amount of time, a process will enter its CS and do some work, thereby making progress.
Now I will explain the last condition, namely bounded waiting. The purpose of this condition is to make sure that every process gets the chance to actually enter its critical section so that no process starves forever. However, please note that neither this condition nor progress guarantees fairness. An implementation of a CS doesn't have to be fair.
Bounded waiting: There exists a bound, or limit, on the number of
times other processes are allowed to enter their critical sections
after a process has made request to enter its critical section and
before that request is granted.
Let's understand this definition sentence by sentence, starting from the last one.
after a process has made request to enter its critical section and
before that request is granted.
In other words, if there is a process that has requested to enter its CS but has not yet entered it. Let's call this process P.
There exists a bound, or limit, on the number of
times other processes are allowed to enter their critical sections
While P is waiting to enter its CS, other processes may be waiting as well and some process is executing in its CS. When it leaves its CS, some other process has to be selected to enter the CS which may or may not be P. Suppose a process other than P was selected. This situation might happen again and again. That is, other processes are getting the chance to enter their CSs but never P. Note that progress is being made, but by other processes, not by P. The problem is that P is not getting the chance to do any work. To prevent starvation, there must be a guarantee that P will eventually enter its CS. For this to happen, the number of times other processes enter their CSs must be limited. In this case, P will definitely get the chance to enter its CS.
I would like to mention that the definition of a CS can be generalized so that at most N processes are executing in their critical sections where N is any positive integer. There are also variants of reader-writer critical sections.
