It is what it claims to be: intensity of light at that wavelength.
However there are a lot of different ways to measure intensity. The "proper" way might be to measure the emitted power in watts into a particular wavelength bin, but that takes a lot of careful algebra and calibration to get right. If your light meter is a digitized CCD readout, however, it probably has an intensity measure for free that has been calibrated at the factory: each pixel on the CCD reports one1 number, where 0 means "no light struck this pixel" and some maximum number2 which means "this pixel received more light than it was able to record"). If you only care about those numbers along one dimension, you can make a plot of them instead of a two-dimensional image.
To show you how straightfoward it really is, here's the example spectrum from anna v's answer. I've taken a row of pixels near the middle of the image (row 60, actually, marked by the arrows) and plotted their red, green, and blue components on a graph. You can see that the blue lines have more blue than the other colors, that the red lines have more red than the other colors, and that the yellow line near pixel 160 is the brightest and is actually saturated (note the flat top and the artifact where the blue "shoulders" are dimmer than the background). You can imagine that I might get a more detailed spectrum graph if I just added up the numbers from all 120 rows of pixels in the image.
Other types of detectors may actually count photons one at a time, in which an axis that's labeled "counts" may actually mean "we counted this many photons in this wavelength bin." You have to read carefully to find what authors mean sometimes.
1 Actually a color CCD will report three numbers, one each for the red, blue, and green sensors nearest a given pixel. You can think of it as giving you three images with the same geometry.
2 If the analog-to-digital converter has $n$ bits of precision the maximum value is $2^n-1$. For 8-, 12-, 16-bit ADCs you get numbers between zero and 255, 4095, 16327.
As for the physics content of your question, each spectral line ideally has three parameters: a location, a height, and a width.
The location of the lines tells you about the energy of the transition involved. Each species of atom has a particular set of energies that electrons are allowed to have, and therefore will have spectral lines whose wavelengths correspond to differences between these energies (as you've indicated that you already know).
The intensity of each line (which is better represented by the area under each peak, rather than the height of the peak) tells you about how common a given transition is. Here is a tool that lets you examine and plot some solar spectral data; you'll see that the absorption lines for hydrogen and helium are very deep, while the absorption lines for other elements are much shallower. That tells you that most sun is made of hydrogen and helium.
If you know that there is a relationship between some sets of spectral lines, you may be able to learn other information from their intensities. For instance the Lyman, Balmer, and Paschen series of hydrogen lines are due to light absorption by hydrogen atoms in their ground state, first excited state, and second excited state, respectively. If you find absorption lines in the Balmer or Paschen series, it means that the temperature of the hydrogen gas is so hot that some of the gas is getting excited from its ground state, then getting excited at least a second time before it has a chance to cool back down to the ground state. By (carefully) comparing the relative intensities of these related line series, you may be able to determine the temperature of the gas. This is how we know, for instance, that parts of the sun's corona are hotter than its photosphere.
Finally each spectral "line" actually has some finite width. Part of that width is always intrinsic to the resolution of the spectrometer, but part is also due to thermal motion of the emitters and absorbers.