"Let's say I have two processes A and B. Assume process A is having a semaphore with it and executing some critical task. Now let us say process B sends a signal to release the semaphore. In this scenario, will process A release the semaphore even if it is executing some critical task?"
One key point to note here is the role of OS kernel. Process B can't send a signal to Process A 'to release the semaphore'. What it can do is request the kernel to give it access to the resource. Process A had requested the kernel and the kernel granted it access to the resource.
Now process A, after it finishes its job, will let the kernel know that it is done with the resource and then kernel grants access to B.
"My doubt arises when a process that does not have the semaphore with it can release the semaphore. What is the use of having a semaphore?"
The key difference between a mutex and a semaphore is, a semaphore serializes access to multiple instances of a resource. Mutex does the same when there is one instance of the resource.
A count is maintained by kernel in case of semaphore and mutex is a special case where the count is 1.
Consider the processes as customers waiting in line at a bank.
The use of semaphore is analogous to the case where there are multiple tellers serving the customers. Usage of mutex is analogous to the case where there is just one teller.
Say there are processes A, B and C that need concurrent access to a resource (lock, file or a data structure in memory, etc.). Further suppose there are 2 instances of the resource. So at most two processes can be granted access at a time.
Process A requests access to an instance of the resource following the required semantics. This request to the kernel involves data structures to identify the resource and maximum number of instances as 2. kernel creates the semaphore with a count of 2, grants A access to the resource and decrements the count to 1, because now only one other process can get access.
Now process B requests access to the resource by following the same semantics. Kernel grants it access and decrements the count to 0.
Now process C requests access, but kernel keeps it in waiting state, because count is 0 and no more than 2 processes can get concurrent access.
Process A is done with the resource and lets kernel know. Kernel notices this and grants access to process C that has been waiting.
In case of mutex, kernel grants access to the resource only one process at a time.