"e-smith" isn't a necessary part of the hostname to access your machine from outside your LAN. To access the machine from outside your LAN, you need either the IP address of the machine, or a hostname which resolves to that IP address. That hostname can be anything. The e-smith comes into play when you're using machines on your LAN.
DNS is pretty complicated, but here's generally how it works:
Every domain has at least two DNS servers, which tell the world whom to ask about that domain. So, when you try to surf to
www.yahoo.com, the 'net asks the DNS server for yahoo.com, which tells you that the IP is, for example, 204.71.200.75. Your computer then connects to that IP address, and everything's fine. When you register a domain, you tell the registrar who the DNS servers are for your domain.
So far, this sounds pretty simple. However, there are two related concepts which complicate the issue. The first issue is caching. You see, my explanation above was oversimplified. If everybody asked ns1.yahoo.com for yahoo.com's IP address, there'd be a lot of unnecessary network traffic, so local DNS servers (at your ISP, for example) cache the DNS records, keeping local copies for the length of time specified in those records.
So, let's recast the example. You still want to go to
www.yahoo.com, so your computer asks your ISP's DNS server who
www.yahoo.com is. That DNS server doesn't know, so it asks ns1.yahoo.com (yahoo's DNS server), gets the answer, gives it to you, and keeps it around for the rest of the day. A few minutes later, your friend wants to go to
www.yahoo.com, so his computer asks the ISP's DNS server. This time, the DNS server knows the answer, so it gives it directly to your friend.
This leads to the second issue, which is propagation. If yahoo.com changed its IP address between your visit and your friend's visit, your ISP won't know about it, because it caches the response for the whole day (actually, for about 11 hours in yahoo's case). The length of time to cache a DNS record is specified in the record itself, but values of one day are pretty common. Consequently, it often takes a day (or more) for changes to a DNS record to spread out to the rest of the 'net.
Now, this has been a brief overview of DNS in general. How does dynamic DNS work? Well, when you register your domain, you give the registrar the DNS servers of a dynamic DNS service. The dynamic DNS service then gives you access to some automated way to update your DNS record, and publishes that DNS record. But, unlike most DNS records, this one is published with a very short lifespan--about one minute. So, your changes will propagate very quickly.
I hope this makes some sense, and helps explain what's going on. If not, let me know, and I'll clarify what I can.
Resources:
Domain registrar for 12 Euros (about US$10) per year:
http://www.gandi.netFree dynamic DNS service, limited domains (only allows subdomains of dyndns.org, mine.nu, etc.):
http://www.dyndns.orgFree DNS service, any domain, limited to 5 domains:
http://www.zoneedit.comFree DNS service, unlimited, but harder to work with:
http://soa.granitecanyon.com