What is the difference between dismissal with notice and without notice?

Your employment contract should state how much notice the employer must give you to bring it to an end. 

This notice must be at least equivalent to the statutory minimum notice, which is a week for each year of service up to a maximum of twelve weeks’ notice after twelve years’ service - unless you behave so badly that the employer is entitled to end the contract immediately – so called “summary” dismissal, for gross misconduct.   

A dismissal with the appropriate notice will be a contractually lawful dismissal. (although it may still be unfair). 

The contract of employment may provide for a payment in lieu of notice, or an employee may voluntarily waive their right to notice and accept payment in lieu.

A dismissal without the appropriate notice is a wrongful dismissal (in other words, it is a breach of contract) unless it is in response to the employee's gross misconduct.

Dismissal without notice (often called 'summary dismissal') in response to gross misconduct should only take place after a proper investigation and disciplinary hearing.

The Acas Code of Practice on Disciplinary and Grievance Procedures (PDF, 167KB) recommends that employers should tell their employees clearly what constitutes gross misconduct.