For each type of analog filters (Butterworth, Chebyshev, etc), there is formulas for computing the order of the filter. Knowing the parameters of the filter frequency response like cutoff frequency and the length of the transition band, you can use these formulas to determine the order. You can see the books like the following for details :

L. P. Huelsman, "Active and passive analog filter design: An introduction", 1993.

hello! i will tell you my way and some things may be useful:

1) look for ripples. If there is ripple only on the pass band, you have chevy 1 and its order is equivalent to the amount of points where the derivative is 0 (on the ripple) . If there is only ripple on the attenuation band, that is chevy 2 and its order is equivalent to the amount of transmission zeroes + the amount of points where the derivative is 0. If there is ripple on both bands, its cauer. Take into account that you might have learnt some transitional filters or some other approximations i am not mentioning, but this is a good starting point.

For flat responses, the aproximations might be butterworth or legendre. Legendre should look like if there was some kind of "steps" while the curve approaches to the transmission band. The order of both should be the attenuation band gradient (slope) divided by 20.

Quite simple:

the attenuation beyond the nominal frequency is

Order * 20 dB/decade resp. Order * 6 dB/octave.

So you wobble the filter with a significant bandwidth and look at the attenuation slope. This is irrespective of the type of the filter (which can be found by the characteristic artefacts in the pass band resp. the stop band. Valid for passive as well as active filters.

See the picture

Ananya - you should not mix the "order" of a filter with the "characteristic (approximation)" of a filter. In the head line you speak about "order" - and in the text you mention "Butterworth". Note: You can have Butterworth, Chebyshev, Legendre,...for all orders (that means: independent on the order).

@Lutz v.W.

I'm that old, still learning... Any other first order filter than Butterworth?

Any other first order filter than Butterworth?

First order is first order. No alternatives possible.

Butterworth (and all other) start for n=2 only.

I should add that n=1 applies to lowpass and high pass filters only - bandpass starts at n=2.

@Lutz

Thanks. Thus no need to 'adjust' my understanding of this topic.

Qualitative comparison of filter magnitudes (LP prototype) for the same 3 dB frequency and the same filter order n;

see too: https://www.researchgate.net/publication/281480600_Theory_of_electronic_circuits

Book Theory of electronic circuits

Qualitative comparison of filter magnitudes (BP) for the same 3 dB frequency and the same filter order n;

More details - see an attachment. 

You can calculate the order of the filter comparing attenuations at two bandwidths that differ twice. 1 order is 6 dB difference, 2 order is 12 dB difference, etc. Start at -10 dB and then look down.