Call-by-push-value is an evaluation strategy that determines when arguments to functions are evaluated. Call-by-value is what every mainstream language does: arguments are evaluated before the function is called. Call-by-name substitutes arguments directly into a function, so they may be evaluated multiple times or not at all. For example, the following pseudocode:

function foo(n, m) {
    sum = 0
    for i in 1 to 4 {
        sum = n + sum
    }
    if false {
        print(m)
    }
    print(sum)
}

foo({print("1"); 2}, {print("3"); 4})

evaluated with Call-by-Value prints:

1
3
8

evaluated with Call-by-Name prints:

1
1
1
1
8

Call-by-push-value combines both by having two "kinds" of parameters: values which are evaluated immediately (call-by-value), and computations which are substituted (call-by-name). So the following code:

function foo(value n, computation m) {
    sum = 0
    for i in 1 to 4 {
        sum = n + sum
    }
    if false {
        print(m)
    }
    print(sum)
}

foo({print("1"); 2}, {print("3"); 4})

would print

1
8

The reason call-by-push-value may be useful is because both call-by-name and call-by-value have their advantages, especially with side-effects. Besides enabling programmers to write both traditional functions and custom loops/conditionals, CBPV is particularly useful for an IR to generate efficient code.

Currently, Scala has syntactic sugar for by-name parameters, and some languages like Kotlin and Swift make zero-argument closure syntax very simple (which does allow custom loops and conditionals, though it's debatable whether this is CBPV). Other languages like Rust and C have macros, which can emulate call-by-name, albeit not ideally (you have hygiene issues and duplicating syntax makes compilation slower). I don't know of any mainstream work on CBPV in the IR side.