It all depends on the specific camera model and the design of the sensor and the firmware installed.

Many Nikon cameras with sensors made by Sony amplify the base signals by 1/3 stop intervals. Other Nikon cameras don't. Even some Sony sensors don't.

As a long time Canon shooter, I can more fully explain how it works with practically all of their sensors produced over the past decade or so.

Most Canon DSLRs set the analog gain at even ISO stops (e.g. ISO 100, 400, 800, etc.).

The +1/3 stops (e.g. ISO 125, 250, 500, etc.) use the nearest even ISO stop and underexpose by 1/3 stop (if you are using an automatic exposure mode such as Av priority or Tv priority), then push exposure 1/3 stop when converting to JPEG. This tends to increase noise in the shadows (because when you underexpose and then push exposure you also amplify the noise) and also reduces dynamic range by 1/3 stop as anything within 1/3 stop of saturated when the photo is taken is blown out when increased 1/3 stop.

The -1/3 stops (e.g. ISO 160, 320, 640, etc.) overexpose by 1/3 stop (if you are using an automatic exposure mode such as Av priority or Tv priority) and then pull the exposure when converting to JPEG. This is pretty much the equivalent of exposing to the right by 1/3 stop: You get slightly better performance in terms of noise in the shadows at the expense of slightly less dynamic range in the highlights.

Even if you are shooting in Manual Exposure Mode and select both the Tv and Av yourself, the camera will include instructions in the RAW file to increase/decrease exposure by 1/3 stop when the RAW file is converted. The exposure meter in the viewfinder when you take the photo will also reflect the 1/3 stop difference. If the meter shows proper exposure for, say, ISO 200, f/5.6, and 1/100 seconds it will show -1/3 stops underexposure for ISO 160, f/5.6, and 1/100 seconds when metering the exact same scene.

For the practical implications of using the +1/3 and -1/3 stops with such a camera, as well as some links to some very interesting test data, please see this answer to Is it really better to shoot at full-stop ISOs?