Firstly, if your website and email are at all important to your business, I would highly recommend you get someone in to look over your setup and document it for you. If you're moving your website to a different machine, you should really understand what your current setup is.
In your question, there are a few different elements that need to be considered:
Domain Name / DNS
Technically these are two separate things, but I'm going to assume you just have a control panel on a web page that lets you change your domain settings including
MX records. The
A record for
example.com points to your website and the
MX record points to your mail server.
The Domain Name settings might be associated with your ISP, or they might be with a completely different company.
This is currently hosted with your ISP.
A record has the IP address for the machine at the ISP that hosts your website. When you move your website to a new host, you will change your
A record to point to the IP address for this new host.
You might also have a
CNAME record for
www.example.com that points to the
A record. You can leave the
CNAME as it is.
This is pointed to by your
MX record. Since it points to your own IP address, I'm guessing you run a mail server on your local network and mail is being delivered there. If that is the case, this should be left as it is.
Again, I would recommend spending some money to get someone to spend a couple of days looking over your network and documenting it for you. It shouldn't be too expensive and will be well worth it in the long term, especially if it prevents your losing your website or access to email for any length of time.