What algorithm does Neatimage use to remove images?

At its most basic, noise reduction typically uses pixel averaging. The problem, of course, is that simple averaging loses detail. Averaging more pixels reduces noise more but loses more detail. Averaging fewer pixels loses less detail but reduces noise less.

Something like NeatImage or Noise Ninja will adaptively adapt to the pixel - for example, it will start by checking for changes that occur with enough pixels that are unlikely to be noise, and where it sees them, averaging over fewer pixels.

They will also consider image channels. A typical digital camera has a filter in front of each sensor. The usual layout is something like grgb (aka, Bayer pattern). Typically, a green filter transmits more light than red or (especially) blue. To maintain color balance in the final image, the brightness of the blues in the image must be "boosted" to compensate. This, however, tends to increase noise in the blue channel. To compensate for this, the squelch will usually do minimal averaging over the green channel, slightly more on the red channel and even more on the blue channel.

The Advanced Squelch usually starts with a noise model for a single sensor and applies noise reduction based on that model. IIRC, NeatImage also allows you to take "dark frames" (like a 30 second exposure with a lens cap) to get a better map of the exact noise characteristics of your precise sensor and take that into account (I know Noise Ninja allows this, and if the memory serves NeatImage, also). Usually, for this to work best, you want to start with something like five dark frames. You statistically analyze them to find 1) which pixels are consistently bright or dark ("stuck pixels") and 2) any consistent patterns you can find in the noise so that you can correct them directly (for example, the part of the sensor next to the processing can become warmer and,hence noisier than other parts) and 3) the type and degree of variation to expect from noise even where there is actually no pattern (for example, some sensors show luminance noise, others mostly color noise).