The Latin expression “lex parsimoniae” literally translates as “Law of parsimony“, or as “Law of economy” or “Law of succinctness” or “Law of briefness” or “law of briefness“, also known as Occam’s razor or Ockham’s razor – attributed to the English philosopher William of Ockham – c. 1285–1348. It was expressed by him in Latin as
entia non sunt multiplicanda praeter necessitate” which translates as
entities must not be multiplied beyond necessity“, or as
more things should not be used than are necessary“.
This problem-solving principle says that entities should not be multiplied without necessity and that for any given phenomenon, in accounting for unexplained facts, the simplest of several competing explanation is always the likeliest to be the correct one.  Although other, more complicated solutions may ultimately prove correct, in the absence of certainty the fewer assumptions made, the better.
However, the ‘simple’ or ‘simplest’ does not necessarily mean the one that is “easier” to explain. It rather means the one with the fewer variables in the equation or fewer types of abstract ideas or fewer guesses or the one with the fewest assumptions. By other words, the Occam’s razor only applies when the simple explanation and complex explanation both work equally well. If a more complex explanation does a better job than a simpler one, then you should use the complex explanation.
In science, Occam’s razor is used as a heuristic (general guiding rule or an observation) to guide scientists.

Some examples

Example I:
Two trees have fallen down during a windy night. Think about these two possible explanations:
The wind has blown them down.
Two meteorites have each taken one tree down and, after striking the trees, hit each other removing any trace of themselves.
Even though both are possible, several other unlikely things would also need to happen for the meteorites to have knocked the trees down, for example: they would have to hit each other and not leave any marks. In addition, meteorites are fairly rare. Since this second explanation needs several assumptions to all be true, it is probably the wrong answer. Occam’s razor tells us the wind blew the trees down, because this is the simplest answer therefore probably the right one.

Example II:
A person is standing on the top of a roof and dropping a feather. In calculating how long it takes for the feather to reach the ground, to make the maths simpler, one might make an assumption: that the effect of air resistance can be ignored. This assumption makes the problem simpler, but is unlikely to lead to a good prediction as to the time it will take for the feather to fall. Thus, making the assumption that air resistance can be ignored is in this case not the “simplest” in concept, but the simplest in other respects (in this case, the maths). Not making the assumption here is the “simplest” in concept because it involves making fewer assumptions.

Example III:
Occam’s razor also comes up in medicine. When there are many explanations for symptoms, the simplest diagnosis is the one to test first. If a child has a runny nose, it probably has the common cold rather than a rare birth defect. Medical students are often told, “When you hear hoof beats, think horses, not zebras”.

Similar sayings
Other thinkers have come up with other versions:
We consider it a good principle to explain the phenomena by the simplest hypothesis possible“. [Ptolemy]
We are to admit no more causes of natural things other than such as are both true and sufficient to explain their appearances. Therefore, to the same natural effects we must, so far as possible, assign the same causes“. [Isaac Newton]
Whenever possible, substitute constructions out of known entities for inferences to unknown entities“. [Bertrand Russel].
Occam’s Razor has also been expressed as the “KISS principle” or “Keep It Simple Stupid!”

And as it usually happens, when studying Ancients Greek Philosophers, Aristotle (384–322 BC) is the first who has written, in his Posterior Analytics (Book A, Chapter 25, download it in Greek here or find it in English here): “We may assume the superiority ceteris paribus [other things being equal] of the demonstration which derives from fewer postulates or hypotheses.”
[Ὅτι μὲν οὖν ἡ καθόλου βελτίων τῆς κατὰ μέρος, τοσαῦθ᾿ ἡμῖν εἰρήσθω· ὅτι δ᾿ ἡ δεικτικὴ τῆς στερητικῆς, ἐντεῦθεν δῆλον. ἔστω γὰρ αὕτη ἡ ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων, ἡ ἐξ ἐλαττόνων αἰτημάτων ἢ ὑποθέσεων ἢ προτάσεων. εἰ γὰρ γνώριμοι ὁμοίως, τὸ θᾶττον γνῶναι διὰ τούτων ὑπάρξει· τοῦτο δ᾿ αἱρετώτερον. λόγος δὲ τῆς προτάσεως, ὅτι βελτίων ἡ ἐξ ἐλαττόνων, καθόλου ὅδε· εἰ γὰρ ὁμοίως εἴη τὸ γνώριμα εἶναι τὰ μέσα, τὰ δὲ πρότερα γνωριμώτερα, ἔστω ἡ μὲν διὰ μέσων ἀπόδειξις τῶν [86b] Β Γ Δ ὅτι τὸ Α τῷ Ε ὑπάρχει, ἡ δὲ διὰ τῶν Ζ Η ὅτι τὸ Α τῷ Ε. ὁμοίως δὴ ἔχει τὸ ὅτι τὸ Α τῷ Δ ὑπάρχει καὶ τὸ Α τῷ Ε. τὸ δ᾿ ὅτι τὸ Α τῷ Δ πρότερον καὶ γνωριμώτερον ἢ ὅτι τὸ Α τῷ Ε· διὰ γὰρ τούτου ἐκεῖνο ἀποδείκνυται, πιστότερον δὲ τὸ δι᾿ οὗ. καὶ ἡ διὰ τῶν ἐλαττόνων ἄρα ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων. ἀμφότεραι μὲν οὖν διά τε ὅρων τριῶν καὶ προτάσεων δύο δείκνυνται, ἀλλ᾿ ἡ μὲν εἶναί τι λαμβάνει, ἡ δὲ καὶ εἶναι καὶ μὴ εἶναί τι· διὰ πλειόνων ἄρα, ὥστε χείρων.]

Perhaps a good relevant statement is Einstein’s quote:
Everything should be made as simple as possible, but not simpler.
The first part of the sentence expresses Occam’s razor, while the second part is a refinement thereof.

Some issues…
The Occam’s Razor is more of a general “rule of thumb”, not a “law”, so the Latin word ‘lex’ is actually a misnomer.
Another issue with Occam’s razor is that the sentence is not really about things (entia = entities), but about explanations or hypotheses.
Besides, the use of the word razor in this context sounds odd to the modern ear, but it is used in philosophy to describe an underlying proposition that permits the ‘shaving off’ of superfluous explanations!

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