Mastery Rank Tests become available when you have enough Mastery Rank (MR) to advance to the next rank. New accounts start at 0 and the (planned) maximum is planned to be 30, though it is not attainable yet. You can increase your MR by leveling up equipment to Rank 30. However, your MR will only increase the first time you level a piece of equipment to 30 – you will not generate additional Mastery Rank for a second copy of a weapon or otherwise re-leveling a weapon through means such as Forma installation. For more information, refer to the Wiki.
A video demonstration of every test is available as a courtesy of MeetTheJoves on YouTube (also featured on the Wiki page). His videos feature a completely new-player friendly demonstration of how to complete each MR test regardless of mods or Warframe abilities.
Increasing your MR will unlock various gameplay elements ranging from Quests to equipment options. For example, Syndicate Primary Weapons require MR12 to use. MR also influences how much Standing you can earn for various Syndicates every daily reset based on the formula 1000 + 1000 * MR = Daily Cap. This applies to faction syndicates such as Arbiters of Hexis, New Loka, etc. It also applies to Conclave Standing and Cephalon Simaris. As of Hotfix 18.13.2, increasing your Mastery Rank level will also improve the base Modding Capacity made available to you on low ranked equipment.