I'm not sure that the obligation cited [above] by Neil, not to exceed the speed limit when driving, is what UK law considers a 'duty'.

What comes to mind in this context is an employer's duty to ensure safe and healthy working conditions for his/her employees [a duty of care], or a teacher's duty to do likewise in his or her classroom in a school, college, etc.

As Neil states, failure to discharge duties effectively can lead to the responsible person or body being liable to prosecution, which can lead to penalties being imposed.

An employer, for example, has a liability to follow established legal procedures when dealing with employees, in regard to conduct, payment of salary, contractual matters, etc.  A trader has a liability to follow relevant provisions in law regarding the sale of products or services, etc.

The employee, or the consumer, has certain legally defined 'rights', which he can enforce in law if he deems that they have been violated.

There is a network of institutions in the UK which deal with matters of this kind :  Employment Tribunals deal with disputes between an employee [or former employee] and his/her employer; local council Trading Standards departments enforce regulations relating to business trade practice; the Health and Safety Executive oversees the enforcement of these matters, etc.

In this context, duties are NOT taxes.  They are NOT any type of пошлина.  Here, duties are legal obligations -  юридические обязанности.  Example:  "Every driver of an automobile has a duty not to exceed the legal speed limit."  

If you breach a duty, you may become legally obligated to pay money.  Perhaps you will have to pay a fine to the government.  Or perhaps you will have to pay damages (возмещать убытки).  Your obligation to pay the fine or damages is called "liability".  

Legal "liablities" can arise even if you do not breach a duty.  A liability can simply be an obligation to pay money.  Example:  You sell me your car for $20,000, and we agree I can pay you on October 31.  I have not breached any duty to you, because October 31 hasn't arrived yet.  However, until I pay you, I have a liability to you in the amount of $20,000.  Another way to understand this is to think of a business.  It has assets and liabilities:  активы и обязательства.

I don't think the example of speed limits being broken/adhered to ... is anything to do with a breach of duty. Not in English law.

Breach of duty is about a faliure to provide the correct standard of care, isn't it?

Thank you very much! Basically I understand what do liabilities mean, the main question I really had was with duties. As far as I understood "duty" is a general idea of following the law with regards to all areas.

And thank you very much for the examples.