Working in the UK
UKCISA is a great source of information covering the different options for working in the UK after your studies.
Targetjobs has a quick overview on visas and permits and the UK Visas and Immigration website explains the whole process in detail.
Pay attention to this guidance from UK Visas and Immigration about switching from Tier 4 to Tier 2 - for students on a Tier 4 visa "You must not leave the UK and make your Tier 2 (General) application from abroad".
Student Circus - a website specifically for international students and graduates looking for work in the UK
TargetJobs has a guide for international students on jobhunting in the UK
Jobs for Graduates has a list of useful websites for finding graduate schemes, jobs and internships
The Recruiting International Graduates Employer Guide (pdf) is a useful document of FAQs for employers about employing international students and graduates. To assist potential employers, make sure that you understand this legislation. It might be helpful to attach this guide to your job application.
Working in Ireland
GradIreland explains how to stay and work in Ireland after graduating.