A site should always be checked on as many different browsers AND OS's (including mobile devices). There are sites out there that will let you know what your site looks like (just visual screenshot) in a ton of different systems (way more than you probably ever know that were out there). There are free and paid services for this. But when possible, check it out yourself. Only have a PC or a MAC, check your local library. The one here has iMacs people can just walk in and use.
Another thing to consider when you check on a computers. Do not assume everyone uses the same screen resultion as you do. This is not the size of the screen in inches, but in terms of how many dots (pixels) across (and down) the screen. Also, keep in mind that not everyone browses at full screen. I mainly design for minimum of 900 wide designs anymore.
Also, a big thing to help pages render the same across different systems (browsers and OS's) is to make sure the code validates, which means it is written in a way all browsers EXPECT it to be. When a page doesn't validate, a browser has to GUESS at what you meant, and while most of the time, they do ok, that is more processing the page has to do to load. This is something to consider on mobile devices.
Long time ago, before things got as complex is they are now (and I'm talking about line 1999) there was an article about how for both Netscape (the king way back when) and IE, more than 50% of the code behind those programs was nothing but figuring out how to guess at displaying broken pages.