Sigma's "Contemporary" lenses are their lower cost, "general consumer" grade.
Their Art and Sport lenses are enhanced in various ways. You can expect Art lenses to have extra attention to image qualities such as background blur and "bokeh", correction of various distortions, better edge to edge sharpness, better flare control, etc. Their Sports lenses will have some of the same plus faster focusing performance, likely will be better sealed for weather resistance and are usuall built sturdier for durability with rough handling.
None of their lenses I'm aware of come in all three varieties.
When it comes to the 150-600mm you have choice Contemporary (approx. $1000) or Sports (approx. $1800).
The Sports version of that lens offers more somewhat refined image quality, faster focusing, better weather sealing and more robust build. It's also considerably more expensive, bigger and heavier.
But if you compare with what powerful telephotos typically cost before these were offered, even the 150-600mm Sport looks like a bargain. Prime 500mm and 600mm lenses commonly cost close to $10,000. Even Sigma's own 500mm f/4.5 still costs $5000. And their 800mm f/5.6 goes for $6600. (Neither of those have OS image stabilization, either... like the 150-600s do.) Sigma's 120-300mm f/2.8 costs $3400, while their 300-800mm goes for $8000 and their massive, 35 lb. 200-500mm f/2.8 costs a stunning $26,000!
Extensive and detailed reviews of both lenses - including comparisons, as well as test shots you can compare side by side and sample images from both - can be seen here:
https://www.the-digital-picture.com/Reviews/Sigma-150-600mm-f-5-6.3-DG-OS-HSM-Contemporary-Lens.aspxhttps://www.the-digital-picture.com/Reviews/Sigma-150-600mm-f-5-6.3-DG-OS-HSM-Sports-Lens.aspxBryan's reviews are very thorough, based on "real world" use as well as various testing and generally quite accurate. Click through "Image Quality" to see test shots done with both at various settings (For sake of comparison, it doesn't matter what camera was used, so long as it's the same format and similar resolution.) You can compare the two zooms side-by-side at various focal lengths and apertures. While all the shots of a lens test target can be informative (found under the Image Quality link), other factors such as vignetting, flare resistance, distortions, and general specifications are also available.
Sigma's two 150-600mm essentially replaced and improved upon three "consumer grade" lenses Sigma previously offered: a 120-400mm OS, 150-500mm OS and a rather hefty 50-500mm OS.