Every festival has its own name that contains in it the essence of what happened then. For example, the word "Pesach" comes from the Hebrew root "pasach". The verb pasach defines Hashem’s action during the tenth plague, as seen in Shemos 12:13 ("I will pass over you" / וּפָסַחְתִּי עֲלֵכֶם) and Shemos 12:27 ("He passed over" / פָּסַח), where the term indicates sparing the Jewish households during the tenth and final plague—the death of the firstborn.
The name of "Purim" which is a Rabbinic festival means what and how does this tie in with the nature of this festival?

The Maharal has a commentary on Megillas Esther called "Ohr Chodosh". His answer comes in his commentary to Esther 9:24.

In Ohr Chadash 9:24, the Maharal of Prague explains that Pur derives from perurim (crumbs), indicating Haman's incorrect belief that the Jews were subject to physical fragmentation, time, and fate. The name Purim signifies that Hashem shattered Haman’s decree into fragments, turning a unified evil plan into harmless "crumbs".
Thus, this name "Purim" adequately describes the nature of this day and its miracles vis-a-vis how the decree of Haman fell apart. Additionally, there is another aspect from the prospective of the Jews. The Purim story took place over many years and what they Jews had to do was to piece them all together.

Every festival has its own mitzvos associated with it. For example, by Pesach it is the eating of matza and not eating of chometz. What are the four mitzvos of Purim and how to easily remember them with sign?

The scriptural source for the four primary mitzvos of Purim is found in the Megillas Esther itself, specifically in 9:22.

To easily memorize these four obligations, Jewish tradition uses a mnemonic device where every single mitzvah begins with the Hebrew letter מ (Mem):