What are the Differences between "Pairs", "Next" and "IPairs"?

They are all iterator functions, used in generic for loops.

ipairs iterates through the integer keys of a Table, starting at 1 and stopping when the next consecutive integer key has a value of nil:

local t = {"gradea", 245, false}
t[6] = "This string won't be reached"

for key, value in ipairs(t) do
    print(key)
    print(value)
end

pairs and next are, effectively, the exact same thing. next is faster, slightly, since pairs uses next internally. They iterate through all members of a Table, but in an undetermined order (between server instances, that is):

local privateKey = {}
local t = {someStringKey = false, [24] = 16, [privateKey] = "swaggy", 1325, false, "anchored"}

for key, value in pairs(t) do
    print(key)
    print(value)
end
--This is, basically, the same as:
for key, value in next, t do
    print(key)
    print(value)
end

These iterators work with Tables. I used 'random' data in my Tables, so sorry if that confused you.

ipairs works, effectively, like this:

local index = 1 --index is another word for 'key'.
local t = {"gradea", 245, false}
t[6] = "This string won't be reached"

while t[index] ~= nil do --~= nil is necessary here because values of `false` are important. `(false ~= nil) == true` Only `nil` Values stop the loop.
    print(t[index])
    index = index + 1
end

That while loop will print the values of t using only consecutive integer keys, starting at 1.

In my Table t, keys 1, 2, 3, and 6 have defined values, so the while loop (and thus, ipairs), will print the values at t[1], t[2], and t[3] and then stop because t[4] == nil.