Not everyone has an IRC client installed on their computer, and for many people that only use IRC occasionally, or may be one-off visitors to IRC, it’s useful to provide alternative access, usually in a form that can be accessed via a web browser.
Java & Flash Clients
Java and Flash both have the potential to connect directly to an IRC server without the web server acting as a middleman. They can be embedded into a webpage, but, unlike gateways, once downloaded, they don’t consume server resources, and they all make seperate connections to IRC servers.
Mibbit is a large, public web gateway designed to be easily embedded as a widget on other pages, as well as used directly. Mibbit is able to connect to most networks, and when used directly, supports multiple network connections and most features of a desktop client, but because it proxies connections to the IRC servers through it’s own service, it’s been banned from some servers and networks at various times.
KiwiIRC is both a open source, self-hosted web IRC client, and operates a public web gateway with support for a number of popular IRC networks. Unlike most web clients, it provides users with a scripting interface, and it’s designed for both desktop and mobile use.
IRCCloud is a commercial web client service, offering free and paid accounts usable via a web browser or via Android and iOS applications. The free plan current is limited to 2 active connections, while the paid plans offer unlimited connections as well as persistent connections with full message history and synchronization across the web application and mobile clients. As of July 2016, they mention BNC as an upcoming service for paid users.
As this is one of the few “hosted” services offering push notifications for mobile users, this is one of the small number of clients for iOS users which can remain connected in the background.
Other Public Web Gateways
Various public web gateways may be usable for your purposes as well, but are generally associated with or restricted to specific networks.