Syncing Office 365 with Existing Domain

Our company had set up an exchange server (accessed via portal.microsoftonline.com) using our domain name (ex *** Email address is removed for privacy ***, and we also had a live website www.our-domain.com). 

We had previously been using Pop Email through godaddy. When we updated to Exchange it was fine, except for the fact you had to change nameservers at GoDaddy which essentially takes down the actual website. 

My question is, how do I use @our-domain.com as an email via Exchange, but also keep our-domain.com live as a website? This error occurred when we transferred our domain and web hosting to another company (FatCow). Our email still sends with the @our-domain group, and we can also still send emails out to other parties. However, we cannot receive emails. 

I need to edit some type of DNS records over at FatCow I believe, but do not know what to edit. I need to keep website live (using FatCow's web hosting/storage), but also connect to exchange. 

Thanks so much.