I am managing my dns using DNS Made Easy. They offer 10 million queries a month.
Does this mean everytime a user comes to my site or an email(using gmail for business configured through DNSME) is sent; it uses up 1 query?
Not necessarily; it means 10 million queries against your authoritative server, which correlates with emails or site visits but doesn't have a direct conversion.
So, if two users are using your site and they share a recursive DNS server, only one query needs to occur to cache a record for its TTL; subsequent queries will get the cached copy without querying your authoritative servers again.
Similarly, an email inbound will tend to use just one query; an outbound email may use several due to
Every one of these records is subject to caching (decreasing query count), and some recursive DNS servers (Google) will query to cache just to keep their cache hot, without getting a query from a client at all (increasing query count).
So, how many queries you'll use will be pretty variable from domain to domain - but 10 million is a pretty healthy quota.
10M queries is a lot. You won't reach this except if your site is very popular.
The queries mainly comes from clients when their computer need to resolve yourdomain.com into its IP address.
Every time a user wants to map your domain name to an IP address to access it, that's a query. Due to DNS caching, not all queries will reach your DNS server. In fact, it should be the case that only a minority of the queries do.
For example, if your timeout value for host names is a day, you shouldn't see more than 1 or 2 queries from Google no matter how many times they forward an email to you because of DNS caching.