This is a difficult one to answer.
Membership of the EU affects member states in an uncountable number of ways, stemming from various EU legislation, regulations, treaties and so forth. Many of these are invisible or only visible in certain situations (like travelling).
To make this more difficult, most of the ways that the UK interacts with the UK is not going to change upon Brexit, because there is a transition period that lasts until the end of this year (this can be extended, though Johnson says he doesn't intend to do that). During that time, the UK and EU will be negotiating a trade arrangement, and that trade arrangement could end up covering some of the things that the UK would otherwise lose after Brexit. Therefore, not all effects will necessarily happen.
Anyway, the headline summary of what changes, either upon Brexit or after the transition period, would be:
- the UK stops paying the EU. The UK's annual net contribution to the EU is about €10.5bn per year.
- the UK stops receiving EU funds. The EU budget is split a number of different ways, and that includes cohesion funds (for things like upgrading infrastructure), funding for things like farmers and fisheries, growth funds, research projects, and administration. The EU spends almost €6.5bn in the UK per year.
- citizens of the UK cease to be EU citizens. EU citizenship is a complex set of perks that make life easier for most actions that involve more than one EU nation. For example, moving between countries, or marrying someone from a different country, or voting in other countries, or obtaining consular protection outside the EU. UK citizens already residing in the EU will keep some of these abilities, but everyone else will lose them. There are similar effects for EU citizens who will not be able to do things like move to the UK as easily.
- trade becomes more difficult. EU regulations greatly simplify intra-EU trade. In most cases, anyone in the EU can trade with another EU country as if the borders between those countries did not exist - no customs declarations, no border inspections, no worrying about goods being rejected because they're different, no difficulty in hiring people from other EU countries to work in yours. This allows companies based in the EU to easily obtain goods, services, people and partners from anywhere within the EU, and has resulted in deeply integrated supply chains and business relationships across the continent. Upon Brexit the UK portion of those links will be disintegrated.
- EU legislation ceases to apply to the UK. The way EU law works is complex and often misunderstood. What happens (in most cases) is that the European Parliament makes law, and then member states individually create their own laws, in their own parliaments, that translate the European law into national law. That means that EU law doesn't apply in the UK because it's EU law passed by the European Parliament, it applies in the UK because it's UK law passed by the UK's parliament. So, the UK now needs to repeal some of those laws, while keeping the ones that it likes. Some of the laws it wants to keep may need to be amended - for example, to replace a reference to an EU regulator to a new UK regulator (since the UK can't use EU regulators any more). The UK has been a member of the EU since 1973. The number of laws that need to be removed, amended or reviewed is immense. This might legitimately be the biggest unforced bureaucratic exercise in all of human history. The UK has had to revive centuries-old procedures to allow ministers the power to change legislation on the fly.
- the EU's free trade agreements with the rest of the world stop applying to the UK. The EU has trade agreements with a large number of non-EU countries, including recent ones with Canada, Japan and South Korea and upcoming ones with Vietnam and Mercosur/Mercosul (a South American trade bloc that includes Brazil and Argentina, among others). Though not anywhere near as comprehensive as the EU single market, these deals still make trade between those countries and the EU much easier and cheaper. Brexit kills the UK's involvement in all those deals. Some smaller countries have already agreed to sign similar trade deals with the UK, but in most cases the deals will have to be renegotiated over the course of years. Membership of the EU means that the UK hasn't needed to do this for decades and trade deals tend to take years to sign (most countries sign maybe one or two per decade) so this is long-term challenge. In theory, trade deals that the UK signs can be customised to the UK's needs (rather than the needs of the entire EU) making them better suited to the UK. In practice, that advantage is likely to be mitigated by the fact that the much bigger size of the total EU market allowed the EU to negotiate favourable deals.
- a trade barrier between Great Britain and Northern Ireland. Trade between GB and NI is currently fully internal trade with no barriers (other than the geographic barrier of the Irish Sea that sits between GB and NI). However, for complex reasons that cannot be easily summarised, Northern Ireland needs to retain fully free travel and trade with Ireland. Since Ireland is in the EU, Northern Ireland needs to maintain regulatory alignment with the EU to do this, and this means checks on trade between GB and NI.
- a million little things stop existing or working. Things like mobile phone roaming between countries, or UK companies being able to buy .eu domain names, or people who have obtained qualifications in the UK having those qualifications recognised automatically in the EU, or UK airlines being able to fly routes between other EU countries, or UK access to EU databases, or being able to easily travel with pets. This would be a never-ending list. The vast majority of changes will go unnoticed by the vast majority of people in the UK. However, almost everyone with any kind of typical level of interaction with society will notice at least some small changes, even if they don't realise that they're due to Brexit.
So that's the set of first-order effects. Those will all have knock-on consequences for many people and businesses. For example, a business that set up in the UK partly because that would allow free trade with all of the EU might consider moving to another EU country. Individuals who fear that Brexit might make their life more difficult might move countries or seek dual nationalities.
