When designing a website, how do I know how users see the colors? I see the colors of the website differently from my Macbook and Windows desktop/notebook. Especially I can’t see light colors on desktop, e.g. gray borders of tables are invisible.
Which one should I care about when designing? Am I able to make all of screens quite similar with a designing solution ?
Short answer: you don’t.
Longer answer: it’s literally impossible to take into account all screen settings and calibrations, circumstances and visual (dis)abilities of viewers. That is why there are guidelines for, for example, contrast of text and it’s background, to guarantee some usability in at least a majority of cases.
My personal workflow includes at least two uncalibrated monitors and some testing on different mobile devices in different lighting situations.
The W3C has made some guidelines about use of colour and necessary contrast which you can read here. Personally, I use this contrast tool for text by Lea Verou to check whether my text colours obey the guidelines or are at least close.