In IC², power tier refers to a range of EU per tick (EU/t) by which blocks and items are grouped, which defines certain ways they behave.
Incorrectly matching power tiers between equipment usually results in either a lack of functionality or a surplus of explosions. (Even if it's only one, any number of unwanted explosions is a surplus!)
Machines (if they are EU acceptors or producers) typically have tooltips including Power Tier: followed by a number, to help you avoid explosions (or trigger them, if you're the HAYO-ish sort).
*As of 2.6.133-ex110, nothing in IC² is labeled as Power Tier: 5.
Any EU acceptor, meaning a machine or EU storage block, can accept up to the maximum EU/t for its power tier without problems. Anything above that usually* causes it to explode, destroying it completely and possibly taking a bit of the surroundings with it. It can accept lower tiers of EU/t, though if it only has a little EU storage (or none, as in the Mass Fabricator) or has several Overclocker Upgrades installed, providing less than the maximum EU/t can cause interruptions. Storage blocks just fill a bit more slowly.
If a machine is compatible with Transformer Upgrades, each one installed will raise the machine's power tier by 1, allowing originally low-tier machines to be directly powered by higher-tier cables, generators, or storage blocks.
*The Creative Generator being a notable exception.
Electric items also have a power tier, though the tooltip does not indicate it (the tier can usually be inferred from the EU storage item used in its recipe).
An item can only be charged by something of the same or higher tier; an Electric Wrench (tier 1) can be charged in anything, but a Nano Saber (tier 3) can only be charged in an MFE or MFSU. (This is also true of EU charging items, and what they can charge.)