EAI is Enterprise Application Integration, which is a fancy way of saying (iirc) the melding together of different enterprise-class applications for business purposes.
I'm hardly an expert, but what I've gleaned from looking at webMethods' website is that they're a company whose products manage EAI between different platforms. They seem to be involved on a high level with web services, and a quick scan over their website would possibly indicate that web services might very well be one of their major ways of leveraging EAI.
Theodore Jonathan Casser
SCJP/SCSNI/SCBCD/SCWCD/SCDJWS/SCMAD/SCEA/MCTS/MCPD... and so many more letters than you can shake a stick at!
Web methods do boast a variety of products that are geared towards verticals(like finance, hr etc) and they also consist services group that solely works on implementing verticals + services for different industries.
ie they make their money with their services experience and product suite geared towards integration of different systems.