Whilst dogs are allowed to bark, they are not allowed to bark so that it causes a nuisance. It is a matter of fact and degree in each case and factors that may be taken into account include the volume, duration of the barking and the time of day it happens. The test is whether it is of a character which makes it intrusive or irritating.
If a complaint is made to a Council, they have a duty to investigate it. The Council may serve a Noise Abatement Notice and if the barking continues then they may prosecute under the Environmental Protection Act 1990. The likely penalty is a fine.
The neighbour may also take action themselves.
Please note that the above summary only relates to the law in England and Wales. You must not rely on it as constituting legal advice and so for specific guidance on your particular doglaw issues please contact us - see our "How we can help" section for details.