8-bit and 16-bit
"8-bit" when referring to image formats typically means that the program assigns 8 bits (eight "1" or "0" values, which together make one byte, capable of representing an integer value from 0  to 255 ) to each pixel's color channel, and each pixel in the files RawTherapee saves has three color channels - red, green and blue.
Most modern raw-capable DSLR cameras use 12- or 14-bit analog-to-digital converters to record the sensor data. This means that when choosing an 8-bit-per-channel output format in your camera, such as JPEG, one loses some information. It's not as simple as it may seem though, cameras record data linearly (due to limitations in hardware design) while JPEG, TIFF and PNG gamma encode their data meaning that they distribute more values in the shadow range and less in the highlights which better matches the eye's sensititvity. This means that an 8 bit JPEG can display as much as log2((1/2^8)^2.2) = 17.6 stops of dynamic range which indeed exceeds the 14 stops of the current best cameras, which explains why you sometimes can see a camera's shadow noise even in an 8 bit JPEG. However due to the fewer values in highlights we lose precision there compared to the camera. Practically this is not a problem when the output file is the definitive one and will not be processed anymore, however a photo can be vastly improved when saved as raw data and processed using a state of the art raw processing program, such as your's truly - RawTherapee.
Once you have processed a photo in RawTherapee, you are faced with the same choice - to save the image with a color resolution of 8 bits per channel, or 16 bits per channel (only TIFF and PNG, not JPEG). If you plan to post-process your photos after RawTherapee in a 16-bit-capable image editing program, it is better to save them in a lossless 16-bit format. Uncompressed TIFF is suggested as an intermediate format, as it is quick to save and stores all the metadata (Exif, IPTC, XMP) of the original file (PNG generally discards metadata!).
There is some confusion over the naming of 8, 16, 24 and 32-bit files. Here is a clarification, but it gets confusing, so put your tin foil hat on. You do not actually need to read this to use RawTherapee, it is just background knowledge. Each of the red, green and blue channels stored in a JPEG, PNG or TIFF file is actually a colorless image, but when you combine these three colorless images together, you get a color image. This is how all digital representation of images works - color images are always decomposed into their components in one way or another. Of the file formats that RawTherapee can save to (JPG, PNG and TIFF), each pixel contains information for three color channels - red, green and blue. We say "8 bits per channel" to make it clear that these 8 bits apply to one color channel only. The reason is that you might encounter references to "8-bit images", and here it gets confusing, because the person who wrote that may have been referring to a grayscale format which stores only one channel, or to a color format that stores three channels, with 8-bit precision each. Another notation for the very same "8-bit" images that RawTherapee saves is "24-bit". Woo, confusing. Or is it? Each pixel is made of 3 channels, and each channel stores 8 bits of data, so we have a total of 24 bits of data per pixel. It gets worse. Image editing programs can also store a fourth channel, called "alpha". To put it simply, alpha describes how transparent a pixel is. These alpha channels also have a "color resolution" of 8 bits. Both PNG and TIFF files can handle alpha, JPEG can't. If you have an 8 bits per channel image with an alpha channel, it can also be described as a 32-bit image; R (8) + G (8) + B (8) + alpha (8) = 32. The ultimate problem is that you can also have an image that assigns as many as 32 bits per color channel. These images can be described as "32-bit" images as well as "96-bit images" (because R (32) + G (32) + B (32) = 96). All real HDR files are stored in image formats that assign at least 16-bit floating-point numbers for each pixel per color channel, such as the EXR format, or 32-bit ones, such as the RGBE format. To summarize, an "8-bit per channel" image with three channels (RGB) can also be called a "24-bit per pixel" image, and a "16-bit per channel" image with three channels can also be called a "48-bit per pixel" image. In both cases use the former (the full "x bit per channel" description, don't just say "x bit"!), it’s more clear what you mean.