What are Euclidean and non-Euclidean geometries?

I read that we had some problems with Euclidean geometry. I was also searching about the elements of a scientific theory. Thus, I decided to study about the Euclidean Geometry, identify the scientific elements in that theory, and know what is the shortcomings with it which lead the non-Euclidean geometry to establish.

First, I was referred to the Wiki page of Euclidean Geometry:

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.

Form the paragraph I had some words which I was not sure about their meanings: attribute, intuitively. I also needed to know the exact definitions for axiom and proposition (theorem). Hence, I looked them up from a dictionary (Longman Dictionary of Contemporary English, 5th edition) [2]:

attribute 1 /əˈtrɪbjət/ verb 2 if people in general attribute a particular statement, painting, piece of music etc to someone, they believe that person said it, painted it etc

intuitive /ɪnˈtuːətɪve/ adjective 1 an intuitive idea is based on a feeling rather than on knowledge or facts SYN instinctive -intuitively

axiom /ˈæksiəm/ noun [uncountable] formal a rule or principle that is generally considered to be true.

proposition /ˌprɑːpəˈzɪʃən/ noun [countable] 5 mathematics technical something that must be proved, or a question to which the answer must be found - used in Geometry

Then, we continue with a beginning explanation on the non-Euclidean geometries and the need to establish such geometry:

For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.... An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

After introductory explanation on these two kinds of geometries, we proceeded into Euclidean geometry in details:

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of axioms. Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions.

 So, I had to understand this paragraph coherently: what could postulate be?

postulate 2 /pɑːstʃələt/ noun [countable] formal something believed to be true, on which an argument or scientific discussion is based. [2]

A more simplified version of the axioms of Euclidean geometry [3]:

1. A straight line segment can be drawn joining any two points.
2. Any straight segment line can be extended indefinitely on a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.
4. If two lines are drawn which intersects a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitable must intersect each other on the side if extended far enough.

Moreover, five common notions are included in The Elements:

1. Things that are equal to the same thing are also equal to one another (Transitive property of equality).
2. If equals are added to equals, then the wholes are equal (Addition property of equality).
3. If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).
4. Things that coincide with one another are equal to one another (Reflexive Property).
5. The whole is greater than the part.

References:
[1]        https://en.wikipedia.org/wiki/Euclidean_geometry
[2]        http://www.longmandictionariesonline.com/
[2]        http://mathworld.wolfram.com/EuclidsPostulates.html