Internaly a computer may store image data in 32 bit format whilst it is being worked upon so that no matter from where in an image you start, data will always be aligned on a word boundary and pixels can be accessed in one operation whereas if the data is stored in 24 bits it will, oddly, require at least two.
Er, that's not what '32 bit' colour means in this case. References to 8-bit, 16-bit and 32-bit colour are all references to bits per channel, as Bifurcator's helpful post explains (the one he gave a link to). As Bifurcator mentions, the number of bits per pixel will depend on the number of channels - which can vary because there may be an alpha channel as well as the other channel or channels. '32 bit colour' is not 8 bits per channel, but 32 bits per channel. The uncompressed file formats that photographers tend to use manage to get an 8-bit-per-channel, 3-channel image into 24 bits per pixel, not 32.
"BTW, how many bits per channel is Adobe RGB?"
Adobe RGB and sRGB are colour spaces, and they can be described by however many bits per channel you wish - there is no inherent bit depth.
Edit
"Which also means that DSLRs that record 12 bits are more than sufficient to cover the full sRGB spectrum?"
Remember that the number of bits the camera 'records' (ie the number of bits coming from the ADC) should not be considered with respect to the output colour space (eg sRGB), but to the sensor's native colour space which may be much greater than sRGB. In addition, the Raw data is a linear, or close to linear, representation of the brightness of the light falling on the pixel rather than the gamma-translated data of sRGB or Adobe RGB. In general you need higher bit depth in a linear description of the brightness of light than in a logarithmic* for the same tonal smoothness and same dynamic range - mostly because we sense light in more of a logarithmic manner than in a linear manner. That can be explained in greater, er, depth if you wish.
*Second edit
In a linear representation, the pixel value is directly related to the brightness. For 8-bit, you have 256 values (0-255). If 100% brightness is 255, 50% brightness is 127. Half of your available values are describing the uppermost stop of your brightness range. This gives great tonal detail in the highlights, but it isn't how we perceive brightness. The famous 18% grey card is about what we think of as 'middle grey', so a perceptual rendering should place 18% at about the half way mark in the data - which is roughly what sRGB and Adobe RGB do. 18% is about two and a half stops down from 100%. Therefore when you go from a linear description of brightness to a more logarithmic one you are stretching out the shadow data and squeezing in the highlight data.
Just as an aside, I've just read 'Understanding Exposure'. It's aimed at beginners, but it makes an error when describing 'black' as 9% and a very bad error when saying that white is 36%. Maybe a question for another thread, if I can be bothered.
Best,
Helen