Broadly, the domicile of an individual is the country that would be considered to the individual’s long term home. It is an international private law concept, but used in the UK’s tax system. By contrast, residence relates to the country or countries in which an individual lives in the short term.
The UK tax system grants tax privileges to UK residents who are not UK domiciled. The tax privileges of non-doms have been reduced in recent years. However, for non-doms who have not been resident for many years, the individual may claim to pay tax on non UK investments only to the extent that the income or gains are brought into the UK.
In both the US and the UK, an LLP is considered to be tax transparent. A UK resident individual would be taxed on his or her share of the underlying profits of the LLP, with credit for US tax.