Web mail is when you launch a browser and go to the website of your email provider and login to check your mail.
Like going to Gmail and logging in. Some advantages of web mail are: ease of use, there is not setup required. No matter where you log on there is all your mail including your sent items. There are very few technical problems that are on your side of the computer when problems come up and your email provider gets the issue solved.
Some downsides to web mail include that if a problem occurs on your email providers servers and their backups do not restore properly there is the risk that you can lose mail you thought you were saving, This is a fairly low risk but does happen. Also if you are not connected to the internet(yes a rarity these days) then you also can not go in and view mail you have already received.
A mail client like Mozilla Thunderbird also have good and bad. The good: You are saving a copy of your mail to your local computer so that no one else can lose your mail, this give you the opportunity to also back up the mail with everything else so you do not lose anything.
You can also check old mail and compose new messages(that will be sent out when you are online again.) when you are not online which can be a great thing for people on the road or who travel a lot in general.Another nice thing about a mail client is that you can set it up to receive mail from sever different addresses. So if you have a few different email accounts you can check them all in once place.
The bad: Setting up web mail can be difficult, it requires you to get settings from your email provider and to enter them in to your mail client to get it to send and receive mail when online. There are also a number of different factors that can determine your settings for your email like the email provider itself, The internet service provider you are currently connected too, If you use security with your connection.
Here is a list of some popular providers settings. Another issue is that if you do not backup your mail if you do have data loss you may never be able to get your mail back.