The general term is color quantization: reducing an image’s many colors to a limited palette of N colors. If the effect is intentionally made to look flatter or more graphic, it’s often called posterization. A stored image using a fixed palette is also described as indexed color.

For your specific task, the workflow is: decode the JPEG to pixel values, then for each pixel choose the nearest color in your palette and replace it with that palette entry or its name. JPEG itself is encoded internally in Y'CbCr, so you don’t work directly on the compressed JPEG data; you first load/decode it through an image library, which gives you RGB-like pixel values to compare.

Useful terms to search: color quantization, indexed color, posterization, and nearest-color palette mapping.