Lasers typically have very narrow peaks in their spectrum, much narrower than 1 nm. It would require quite a high-resolution spectrometer to measure such a peak precisely. But if I simply need to find out order of magnitude of spectral line width, is there any way to do this without resorting to high-resolution/expensive instruments?
To measure laser line width, you can build a Michaelson interferometer with the length of one beam path adjustable. Measure the interference fringe contrast as a function of path length difference. The line width is inversely related to the range of path length difference over which you can see the fringes. So, a single-frequency laser will produce fringes over several meters of path length difference, while a cheap HeNe laser will produce visible fringes only over about 30 centimeters of path length difference. That range of path length difference over which high contrast fringes are obtained is called the coherence length. Early violet diode lasers had only millimeter or so of coherence length. Now it is possible to buy diode lasers with many centimeters of coherence length.
Edit: Sometimes it is mistakenly stated that coherence length is equal to or dependent on the cavity length. That is mostly nonsense. A laser cavity of any fixed length can support only certain oscillation modes, the lasing medium will amplify only a finite range of wavelengths, and the cavity mirrors will only reflect a certain range of wavelengths efficiently. All those factors together limit the oscillation modes that contribute to the output beam. The more modes there are in the output, the wider the line width of the laser; and the wider the line width, the shorter the coherence length. If only one oscillation mode contributes to the output, the laser is single-frequency and has a very long coherence length: often on the order of ten meters or more.
One more thing can limit the number of modes that contribute to the output beam: an etalon in the cavity. An etalon is essentially a flat plate of glass. The etalon itself will transmit only certain wavelengths of light efficiently, depending on the thickness of the plate and the angle at which the beam is incident. When an etalon is in the laser cavity and tuned (tilted) precisely to one of the "natural" modes of the laser, only that mode will be amplified and the laser will output a single frequency with many meters of coherence length. In a diode laser with a high-finesse cavity, the cavity length can be so short that only one optical mode is supported. In this case, the diode laser acts as its own etalon and outputs a single-frequency beam.
Sometimes it is stated that the cause of short coherence length in a laser is "random phase shifts". This is sort of true, but mostly mistaken. If it were possible to measure the instantaneous phase of a laser beam with finite coherence length, it's true that the phase would vary rapidly. It would even appear that the variation is random. However, the phase variation would be due to the presence of multiple superimposed wavelength/frequency components (modes) in the beam. In essence the phase variation would occur at a mix of frequencies corresponding to the frequency differences between the different modes output by the laser. It is far from random. Of course, anything that alters the mode structure of the laser- such as temperature drift- will slowly change the frequency content of the laser's output beam.
Bottom line: measure the coherence length of your laser, and that will give you a handle on the line width. See this website for the formula to calculate line width from coherence length.