Does this mean it is a (mixture) sample injected into a MS without the preceding chromatographic separation (as is typical in, say, a GC-MS or an LC-MS)?
No. A chromatogram is the measurement of something after separation by chromatography. So you got chromatography beforehand. In case of direct injection into MS the term is usually used too, if you record over time like you would do using chromatography.
What's the point in doing this?
Ok so let's imagine I'm running a sample of an unknown mixture and I'm recording m/z 100-1000. Now I don't know at all what to expect, so selecting randomly a specific mass for my chromatogram would, most likely, lead to nothing. What we can do is just integrate over the whole mass range and take the total number of all ions, so we see a peak this means there are more ions than for baseline noise. That should correspond to a substance reaching the detector.
Wouldn't the resultant spectrum be very noisy since it's essentially the sum over all components of a mixture of unknown proportions.
Depending on the method: yes that might be a problem. And if the noise is very high then even a strong signal of a single m/z wouldn't really contribute to the total number of ions, so we won't see a peak in the TIC.
A way to circumvent this problem is using a base peak chromatogram. Base peak is the highest peak in your spectrum, and you are plotting the intensity of the highest peak over time. Now for noise there might be a large amount of total ions, but the intensity of each m/z is usually small. If you got a specific signal from a substance you will get much higher signals there, so the base peak chromatogram will show a peak.
What's the alternative?
Collecting data in TIC mode is an alternative to collecting data in SIM mode. SIM stands for "single ion monitoring" or "selected ion monitoring". Let's say that you know your sample contains acetophenone, and the four largest peaks for acetophenone are at 120 (molecular ion), 105 (base), 77, and 51 m/z. Instead of integrating over all ions, you could monitor only those masses (or only one of them). You would know where the acetophenone eluted and could quantify it relative to a standard, but you would know little about the rest of the sample.
By analogy, consider a LC system with a UV/Vis photodiode array detector. The detector can monitor single wavelengths or it can capture a full absorbance spectrum at each time interval. A mass spectrometer collecting data in TIC mode is like a PDA detector collecting a full spectrum. A mass spectrometer collecting data in SIM mode is like a PDA monitoring individual wavelengths.