Web-Based Clients can be very handy. You don't need to worry about installing anything on your device, or worry about extensive configuration and connection issues. You generally sign up, log in, select your network and channel, and start chatting.
Undernet Webchat – Undernet Connect
The Undernet network has it's own web-based client, Undernet Connect, at https://chat.undernet.org/. It could not be easier. You enter your nickname and the intended channel, and click Start. Too easy! You can also use Undernet's log in and host masking service. Just check the Use Services box, fill in the username and password, and you are all set. This will result in your address looking like this: YourNick!webchat@yourusername.users.undernet.org. Find out how to register at Protect Yourself on IRC. Using Undernet's Services will effectively prevent others from seeing your current IP/Host Address.
If you are not using the Services, Undernet Connect uses your current IP/Host address as follows: YourNick!webchat@your-ip-address.or.host.address. If you are concerned with privacy, use a VPN.
IRCCloud.com
IRCCloud has a more robust set of features that Mibbit, and it works in essentially the same way. One nice feature is the ability to copy things and save them in your account at IRCCloud directly from the chat windows. This includes screenshots. IRCCloud is better for privacy, as it does not directly expose your current IP or Host Address.
Note: the following services create an Ident for you, using your current IP/Host address. It's a simple hex that will look something like this: YourNick!ae55fdee@irc1.mibbit.com (in the case of Mibbit). This means that anyone can reverse that hexed IP and get your real connection information. Use a VPN! If you are connecting to the Undernet, you can also log in and set your mode +x, giving you an undernet host mask. No one can see or determine your current IP or hostname (though IRCOps and IRCAdmins can see it).
Mibbit.com
Mibbit is a web-browser based IRC client. You register and log in, select your IRC network, set up your starting nickname, and click Connect. Once connected, you can join any existing channel, or create your own. Or both! Mibbit has a good list of features, and it easy to use. However, Mibbit will directly expose your current IP or Host address in the WHOIS ‘Realname' (GECOS) field. Use a VPN if you are concerned about privacy.
Kiwiirc.com
Kiwi IRC is very similar to Mibbit.com. It has less features than both Mibbit or IRCCloud, and is extremely easy to use. It also does not directly expose your IP or Host Address, but it does create your ident using a hex of your current IP. Use a VPN!