Unfair Dismissal
There is a statutory cap on the amount of compensation which a Tribunal can award for a successful unfair dismissal.  The current amount is £78,400 and is updated each year. 

Employees generally have to have been continuously employed for one year to allow them to claim unfair dismissal. There are exceptions, including for example dismissals for reasons connected to pregnancy or childbirth, health and safety activities, whistleblowing exercising various time off rights, or asserting a statutory right under the Employment Rights (Northern Ireland) Order 1996.
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There are five potentially fair reasons for dismissal: capability or qualifications, conduct, redundancy, breach of a statutory duty or restriction and "some other substantial reason".  It is the employer's responsibility to prove the an employee’s dismissal is justified.

Employers should follow 3 step procedure when dismissing an employee – a failure to do so may result in any compensation awarded being increased by up to 50% and further result in a finding of automatic unfair dismissal. 
Guidance: https://www.lra.org.uk/images/publications/44301_final_low_res_web_-4.pdf

There is a three month time limit in which to submit a claim of unfair dismissal to an Industrial Tribunal.  This time limit is extremely strictly adhered to and it is exceptionally rare that a claim will be accepted after the time limit.  The three months start to run from the “effective date of termination” ie the date the employee is informed/they become aware that they are being dismissed.
It is a common misconception that it you have submitted an appeal against your dismissal that the time limit has not yet started to run – this is incorrect.  Time starts to run from the effective date of termination, regardless of whether there is an appeal pending or not.  Therefore,  advice should be sought promptly and claims should be made to the Tribunal without delay.