It's easy to find examples of equivalence relations (for example, A shares room with B), but I can't seem to find a real life example of an order relation (that is, a relation that's reflexive, antisymmetric and transitive).
In particular, I can't seem to find a (real life) relation that is reflexive, yet not symmetric.
On
An order relation that is used in schools around the world every day: (non-strict) alphabetical order. Define $w_1\preceq w_2$ to mean "either $w_1$ is the same as $w_2$, or $w_1$ comes before $w_2$ in alphabetical order".
Of course we should recognise that it's not altogether easy to give a precise definition of alphabetical order. For example, do capital letters come before or after lowercase? Or do they count as the same? And what about punctuation: does "its" come before "it's"? However these are really linguistic problems rather than mathematical problems, and as long as we can sort out what it actually means, alphabetical order is definitely an example of a partial order.
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Most simple corporate organizational charts, where every person has at most a single manager, can be seen as an order. Then we say $A\leq B$ if $B$ is some number of steps above $A$, including $0$ steps above.
This example has the advantage of not being "linear-ish."
On
A relation over geographic regions:
New York State is within the bounds of the United States.
New York City is within the bounds of New York State.
Manhattan is within the bounds of New York City.
Manhattan is within the bounds of Manhattan; where else would it be? Because it is within New York City, it must be within the bounds of New York State, and therefore also within the bounds of the United States.
On the other hand, since New York City and New York State are two different things, not two names for the same thing, the above implies that New York State cannot possibly be within the bounds of New York City.
It is possible for a region to be within the bounds of two other regions, neither of which is within the bounds of the other, but that doesn't violate either the reflexive or transitive property.
Daoud is not richer than Fatma.