I just use hotmail and get my mail both on my computer and iPad with no problems.
Hotmail uses Exchange servers (last I checked). Similar to IMAP servers, all your email is synced between the server and your email client(s). This means that the server and all your email clients always look the same (once they've had time to sync with the server, which should aways be the most up-to-date).
More info below, for the curious.
POP3 copies email from the server to your email client, and then either leaves or deletes it on the server depending on the email client's settings. If you only have one email client/device, then this works fairly well. If you read your email on more than one email client or device, the results vary depeding on each email client's settings (download and delete, or download and keep), and when you last used which client.
It is also up to each email client to remember what emails it already downloaded. If for some reason the client forgets this, then it will download all email from the server. When lots of old emails suddenly reapear as new/unread, this is usually the cause.
POP3 is based on an older POP email service, when most people only had one computer and one place to read their email. It has adapted poorly to the new mobile world, but a lot of ISP (Internte Service Providers) have kept it because maintaining POP servers is cheaper.
IMAP is a newer standard that has translated well to the mobile world. Pretty much everyone that offeres independent email services uses IMAP; with the exception of businesses that use Microsoft Windows/Exchange servers, and Microsoft themselves.
Web based email may or may not use these services. It accesses the same database of emails, but because it's served via a webpage the site can use pretty much any software they want. There is no need to adhere to a standard. Because a website is a single access point you never have to worry about email showing up on one device and not another. Of course, there is no offline access.
Most email providers offer web service in parrallel to IMAP, Exchange, and/or POP3. When there is something you can't do on your email client, or something does not look right, it's a good idea to check the webmail client.