EU citizens and their families currently have the right freely to travel, live and work across the EU. When the UK leaves the EU then this could have wide ranging implications, as UK citizens and EU citizens of other member states will no longer enjoy this automatic right of free movement to travel, live and work in EU/EEA member states. This restriction on movement will have a significant impact on current and future labour and trade markets in the UK and the EU and global trade outside of the EU.
When the UK leaves the EU it can impose its own controls on who is admitted to the UK and under what circumstances. EU nationals could be subject to the same visa rules and requirements that currently apply to non-EU nationals. It has been suggested that we may adopt a points-based system, similar to Australia, however this is unlikely to reduce net migration significantly. If it is no longer bound by free movement law, the UK may have more powers to deter and restrict the presence of certain categories of EU migrants and more powers to remove/deport national immigration offenders. The flip side is of course that while the UK could impose its own controls on EU immigration, the rights of UK citizens to travel, move to or work in an EU member state will be restricted by any visa entry requirements or restrictions that EU member states choose to impose.
Brexit is unlikely to result in any immediate changes as there will be a complex and lengthy transitional period. The UK will need to negotiate and agree on arrangements for its future relationship with the EU. Some EU nationals may have acquired rights to stay under UK legislation. For others, there could be reciprocal arrangements agreed between other EU countries so that EU nationals already working in the UK may be granted permission to stay in the UK and UK citizens working in other EU countries may be granted similar permission to stay in return.