People used different numeric systems which didn't have zeros - or, indeed, any of what we use as numbers. Our numbers are Arabic in origin - Roman numerals were used in Europe before them, up to about the 13th century - and well beyond in many places.

Roman numerals count 1-10 as follows: I, II, III, IV, V, VI, VII, VIII IX, X. They're read, approximately, by finding the highest value 'numerals' from I (1), V (5), X (10), L (50), C (100), D (500) and M (1000), adding the numerals to the right, and subtracting those to the left. This becomes more complex as you arrive at something like MDCCCXIX (1819), because the IX at the end (9) is a separate sequence.

This system of numbering doesn't have concept of 'zero' at all; there is no symbol to indicate 'nothing'. It also doesn't use the same concept of the position of the numeral increasing its magnitude (that is, '1' has a different value, more or less, in 1, 10, and 100). However, zero isn't strictly necessary in such a number system either; we could use any set of 10 symbols for our numbers and they'd work just as well. It's only when you start to have 'nothing' as a concept, and then negative numbers, that zero is actually significant.

Sources:

Leo Corry, A Brief History of Numbers