Often programmers tend to favour a particular type of browsers. This is fine as long as when they are checking the site they do not neglect to check how the site looks and functions on other popular browsers. Internet Explorer, Firefox, Chome, Safari and Opera are amongst the most popular and even within theses browsers you have multiple versions. Unfortunately all browsers render some elements of code differently so you will need to write a website that is able to function on all. There are so many different versions of the same browsers that it is impossible to get a site to look exactly the same on all. Normally a programmer will tell you what browsers they will test the site on but if you are unsure always ask. If you include flash in your web design, you will have to be aware that some browsers will not display this or that the user may not have Adobe Flash installed.
These are all pointe you will need to discuss with your designer and programmer upfront to ensure all parties are happy with the end result of the website.