Two numbers do most of the work, and for mainstream pairings neither will trip you up. It is at the demanding end that amp choice starts deciding the outcome.
Sensitivity, in dB, tells you how loud a speaker plays for a given input, usually quoted at 1 watt at 1 metre. Higher is easier to drive. The consequence is not academic: every 3 dB less sensitivity means the amp has to deliver double the power for the same volume, so an 85 dB speaker in a large room is asking a lot more of your AVR than an 90 dB one at the same seat.
Impedance, in ohms, tells you how hard the amp has to work to push current into the speaker. Nominal 8 ohm loads are easy. Nominal 4 ohm loads, and speakers whose impedance dips low in the bass, demand real current and deserve a stronger amp section. Ask a modest AVR for that with all channels driven at reference level in a big hall and it compresses, runs hot and eventually goes into protection. Nothing breaks, but the film gets smaller.
Where do you find the numbers? On the manufacturer's published spec page for the speaker, and in the AVR maker's supported-impedance guidance. Look at both, and be a little sceptical of the sensitivity figure, since methods vary between brands.
Rule of thumb: mainstream speakers and mainstream AVRs from authorised ranges are designed to work together. An 8 ohm, 88 dB bookshelf pair on a mid-range AVR in a 12 by 15 foot room has no problem. The picture changes with big rooms, long listening distances, low-impedance audiophile speakers and reference volumes. There the amp section, not the feature list, decides whether the system sounds effortless or strained, and it is where a pre-out to an external power amp starts to earn its place.
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