While being a lossy format, JPEG files can be easily compressed into small file sizes. It's optimized for photographs and other images which contain many colors, and/or large runs of very similar colors. Programs like Adobe Photoshop and Paint Shop Pro allow you to choose the level of compression, giving you somewhat of a choice between image quality and file size.
BMP files are a lossless and uncompressed format invented initially by Microsoft. They can be quite large, but do not lose any data. Due to the file size involved, it's unfeasible to use BMP’s for most tasks, especially considering that there are lossless formats out there that can be compressed into much smaller file sizes.
This is also a lossless format. It works much differently than your common TIFF files, or any other image file types really. It looks for patterns in the image that can be used to compress file size. The compression is reversible, so converting to other formats from PNG will not result in any loss of image data, provided that the format you are converting to is not a lossy format.