In the axiomatic approach, one defines the plane as anything that satisfies a given list of properties. These properties are called axioms. The axiomatic system for the theory is like the rules for a game. Once the axiom system is fixed, a statement is considered to be true if it follows from the axioms and nothing else is considered to be true.
The formulations of the first axioms were not rigorous at all. For example, Euclid described a line as breadthless length and a straight line as a line that lies evenly with the points on itself. On the other hand, these formulations were sufficiently clear, so that one mathematician could understand the other.
The best way to understand an axiomatic system is to make one by yourself. Look around and choose a physical model of the Euclidean plane; imagine an infinite and perfect surface of a chalkboard. Now try to collect the key observations about this model. Assume for now that we have intuitive understanding of such notions as line and point.
(i) We can measure distances between points.
(ii) We can draw a unique line that passes thru two given points.
(iii) We can measure angles.
(iv) If we rotate or shift we will not see the difference.
(v) If we change scale we will not see the difference.
These observations are good to start with. Further we will develop the language to reformulate them rigorously.