I create a logo with a circle and because it has transparent area, I used png. The thing is that on my website it doesn't look smooth. There is a lot of aliasing... Should I use svg for logo? Is it well supported by browser?
I would have thought any problems you're having with the .png file are related to the original image itself and the quality of its conversion to .png, rather than a problem with the .png format.
In answer to your question though, Safari, Firefox, Internet Explorer, Edge, Chrome, and Opera all offer SVG rendering support.
If it was me, though, I would first try to see if you can get it to look as you want in PNG format first, only because PNG is more widely supported generally.
Your specific example's lack of smoothness is not because png renders bad. It is just because you did not exported your png file in a correct way.
If you have your original logo in vectors turn anti-alias on. In some rare cases you can export it two times larger and resample it on Photoshop or Gimp. If you do not have the original in vectors... you need to draw it again anyway. (Inserting a bitmap inside Ilustrator wil not magicly convert it to vectors)
For something like a Logo, you can use either format, but I would only use SVG if I really need to scale an image. Some map, some blueprints, some diagrams.