The Greeks for the most part did not use our modern algebraic notation, which is much newer. For much of the history of mathematics, an equation we write as x^2 - 7x - 2 = 0 would be something like "Determine the quantity whose square is equal to seven of itself and two units." (Notice the avoidance of negative numbers: They would have first rewritten this as x^2 = 7x +2.) Diophantus' Arithmetic, a work on what we would call finding integer solutions of polynomial equations, had some symbolic notation for these equations, but it was pretty limited compared to modern notation.

Despite this, in some ways the Elements looks very much like a modern geometry textbook: there are diagrams with points labelled by letters, lines called AB and angles ABC (or ABΓ) and statements and proofs of theorems. The difference is that there is no symbolic algebraic notation. There are also no coordinates; points are not represented by pairs (or triples) of numbers. While we might state the Pythagorean theorem as "the squared distance from the origin to the point (a,b) is a^2 + b^(2)", Euclid would say that the sum of the (areas of the) squares on the two shorter sides is equal to the square on the hypotenuse. (Compare the diagram here.) Things like the Law of Cosines get more complicated to state without algebra, but it's still possible.

A lot of this came down to the Greek conception of numbers. While we freely talk about not only rational (2/3) but also irrational (the square root of 3) and even transcendental numbers (π, e) in elementary math education, you'll notice that there isn't a good, general definition of what a real number is. This is because the definition is actually pretty tricky, and because it only becomes really important when you start doing rigorous calculus, which is something normally only taught to mathematics majors.

The Greek mathematicians were quite concerned about rigorous proofs and definitions, so they stuck to what they knew. They were ok with whole numbers (1,2,3...), and they were ok with length and areas, but generally speaking those were different things. They would not say "Line AB has a length of 2," but instead "Line CD is twice the length of AB." This idea is retained somewhat in expressions like "Line AB has a length of 2 units;" the "unit" is some fixed length we are using for comparison.

Similarly, the Greek mathematicians did not prove that the square root of two is an irrational number, they proved that the ratio of the diagonal of a square to its side is not expressible as a ratio of whole numbers. The point of stating things this way is that you don't have to worry about what an "irrational number" is, but instead rely on geometric concepts like length. Another statement is "the ratio of the area of a circle to the square on its radius is equal to the ratio of a circle to its diameter," which we write as area = πr^(2). π was defined to be the ratio of a circle's circumference to its diameter, and it was a theorem that this was the same for all circles.

You shouldn't think of this as "primitive" or "inferior:" the Greek mathematicians had a rigorous concept of quantity that worked for their purposes. The Elements has sections defining exactly what it means for lengths or proportions to the same that closely resemble what we now call Dedekind cuts, which are a way of constructing real numbers. Archimedes even figured out how to approximate irrational ratios like π using rational ones, by using inscribed and circumscribed polygons.

Nonetheless, there were some problems with this approach. One objection to early algebraic notation in early modern Europe was drawn from Greek ideas about quantity: things like x^2 + x don't make sense, because you're adding an area to a length. Physics students might recognize this as a dimensionality issue, and the solution was similar to the way we deal with it, namely an implicit or explicit coefficient for x with units of length. There were a number of related conceptual issues that had to be worked out to make sense of algebraic notation.

Sources: The History of the Calculus by Carl B. Boyer. This was the textbook for an undergraduate mathematics class I took on the history of calculus, but I've now realized it's probably rather dated. If a real expert shows up I defer to them.