filter order
Filter order is, loosely, how many energy-storing parts (capacitors and inductors) shape the response, and it tells you how steep and selective the filter is. A first-order filter is the gentlest possible; each step up in order makes the cutoff sharper and the stopband deeper, at the cost of more parts and more side effects. Order is the single dial that trades complexity for steepness.
Concretely, the order equals the number of reactive elements that act, and the asymptotic roll-off is 6 dB per octave times the order (20 dB per decade times the order). A first-order section, one R and one C, rolls off at 6 dB per octave. A second-order section, the basic building block of serious filters, rolls off at 12 dB per octave and can have a controllable Q. To build higher even orders you cascade second-order sections; an odd order adds one first-order section. So a 4th-order filter is two second-order sections in series (4 times 6 = 24 dB per octave), and a 6th-order is three.
You pick the order to meet a spec: given how much attenuation you need at a given frequency, the order is whatever makes the roll-off steep enough to get there. The honest caveat is that higher order is not free. Each added section adds cost, board space, and op-amp noise, and crucially it makes the filter more sensitive, component tolerances matter more, and the phase response gets worse. A real design uses the lowest order that meets the spec, not the highest you can build.
Spec: a low-pass with 1 kHz cutoff that is at least 40 dB down at 4 kHz (two octaves up). A first-order filter gives only 12 dB there. You need order n such that n times 6 dB per octave times 2 octaves is at least 40 dB, so n times 12 is at least 40, meaning n equals 4. A 4th-order filter (two cascaded second-order sections) meets it.
Order is chosen to hit a required attenuation; round up to the nearest order that meets the spec.
Higher order means steeper roll-off but also more parts, more noise, greater component-tolerance sensitivity, and worse phase response. Use the lowest order that meets the spec, not the highest you can build.