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A filter may have only feed-forward coefficients (FIR filters) or also feed-back coefficients (IIR Filters). Each of these coefficients is actually a gain value multiplied to a delayed input sample.
For FIR filters, the maximum number of samples by which an input sample is delayed is the order of the FIR filter. This is the order of the feed-forward part of the FIR filter. The feedback part in an FIR filter has zero order.
For IIR filters, there can be non-zero sample delays in the feed-forward as well as the feed-back signal chains. The larger of these two numbers if the order of the IIR filter.
If so, you should examine the polynomial order of the FIR/IIR filter.
For example, a filter usually can be expressed as the following form:
H(z) = A0 + A1 * z-1 + A2 * z-2 + ... + An * z-n
where Y(z) = X(z)H(z).
If n is finite, such as n = N, then we can confirm it is a FIR filter, and sure the order of the FIR filter is equal to N.
Of course, the delay of sample equals N undoubtedly, because of X(z)Z-N => x[n-N] through inverse Z transform.
Meanwhile, you should be enlightened by the idea that the same equation can be expressed as an IIR filter, if n goes to infinite.
For example, H(z) = 1/(1-z-1), this can generate infinite polynomial terms if you do the division, the response of the filter cannot stop if you give an impulse to this filter.
That is why we call it IIR (Infinite Impulse Response), and (1-z-1) is the feedback term as axebomb mentioned.
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