Currency:
Firstly, people saying that the pound has recovered against the Euro don't understand the currency markets. Certain economies are linked, so (for example) when Sterling tanks that affects Ireland and our closest neighbours with whom we do the most trade. The Irish stock market fell off a cliff when the Brexit was confirmed. As a result, even though we are not in EMU, a tanking pound will, at least in the short term take the Euro down with it. To escape the falling pound investors would flee to the Dollar, and perhaps in a lesser way some of the more stable Asian currencies, so to see where the pound really is one needs to look at the dollar or the Yen. It's bad, and that isn't changing any time soon unless there is an even bigger crisis somewhere else. Although, even if a big US crisis (e.g. Trump being president) would make the comparison exchange rate LOOK better, in reality we wouldn't have improved, they would have sunk, and that means the whole world would feel the pain as in 2008.
A weakened pound is good for exporters, but the reality is that we are completely reliant on imports in our day to day lives - the whole of the UK doesn't produce enough food to feed London - so as well as the cost of travelling shooting up, so does the cost of day to day life, and if it happens in a recession, things get more expensive at the time we can least afford it.
Stocks:
Large companies, especially those likely to be directly affected by any changes to free trade/movement of people will be hit by the uncertainty and probably issue profit warnings damaging the value of their stock. Financial institutions based in the UK are certified by EU regulations, so many of these will possibly have to move some/all their workforce away from the UK. Sudden relocation is a huge cost to a company affecting profits, so the uncertainty will cause some investors to cut their losses and sell off these stocks, which is why some of the sectors seeing the biggest collapses in share price were finance and aviation. There will be others too, but I'm not an equities analyst, and you would need to go through all the losses and gains to see the patterns and try to work out why and whether they are likely to be short or long term losses.
Other than for individuals with share portfolios, for ordinary people in the UK falling stock prices means a whack wiped out of pension pots which in the longer term means later retirement and lower pension incomes. When people say about yesterdays stock market crash that 'it recovered' they need to remember that interest rates are low, so maybe you would hope for steady growth of a few percent per year from a fund...even with the small bounce back, in a single day the equivalent of a full year's growth and more can be wiped out and probably was. Any equities people on here might be able to give you more detail, but in layman terms, anything that affects the profitability of big companies is bad for share prices and pensions.
House Prices:
If big industry such as finance moves then highly paid jobs leave the economy. The people who pay most of the tax aren't super rich semi-resident billionaires, they are highly paid (overpaid?!) PAYE workers in high earning sectors such as finance. These are people who are likely to move to follow their jobs. Apart from the hit to the tax take, these are people who own/rent substantial houses and the laws of supply and demand means that you would expect the prices in this housing sector to take a hit, and this will trickle down and up. An educated guess would expect a fall in house prices which will be great for people trying to get on the ladder, and horrific for those trapped in negative equity. Again, it's very much a wait and see, because until the negotiations are complete nobody really knows anything....although there has already some stagnation and even price falls in this sector as people waited for the result of the referendum. A housing crisis also affects the a wider industry - architects, solicitors, builders as was seen in the Irish property crash in recent years.
Restrictions in free movement and/or a recession hitting jobs might also cause lower paid migrants to move on to other EU countries with better opportunities. Again, falling rents (good for renters, bad for those paying mortgages on rental properties) and lower house prices at entry level housing...good for some, terrible for others. The killer about recessions is that they don't affect everyone equally. House prices in certain parts of the country have been out of control and have probably needed some governmental input to take the heat out of the market for a long time. The difference is that when the adjustment comes about in the less controlled way caused by a recession the effects cannot be controlled and spread out gradually. So as in the Irish property crash I expect some people's lives will be completely destroyed - people already committed to the property market find themselves a decade later raising families in tiny one bedroom flats with negative equity mortgages and childcare bills requiring two professional incomes to service, while others 1-2 decades younger can get onto the housing ladder at half the price with ease. In between was a whole generation of graduates and skilled workers who were forced to emigrate and build lives elsewhere. Many will never return.
General Recession:
Our economy is entirely dependent on growth...in real terms, Western life (where we all live way beyond the means of what we actually produce) is a Ponzi scheme and the working population needs to consistently grow so each generation can fund the retirement of the generation ahead of them. It helps if the growth is in high earning sectors. If we just keep the workforce at the same size we cannot afford to live as we do. When growth slows (or falls back) it is a double whammy - lost jobs mean that at the same time as tax take income falls, the state has to support more jobless people with benefits. One way of addressing this is to extend the retirement age, another is to cut services (NHS, education, council services), benefits and pensions. Again, all this will take time to occur because of the negotiation period and then the time it takes for the damage to seep through the economy, but it makes it particularly ironic that the largest groups of 'leave' voters were those reliant on benefits and pensioners...
Political Instability:
Scotland could devolve.
Less likely (but possible) is devolution by NI because a new border will have a really negative impact on the daily lives of many who live lives which straddle it. This could trigger bids for independence by the Basques, the Catalans and other European regions, again destabilising Europe while settlements are negotiated and agreed.
Putin is clearly seeking to destabilise the EU - he will doubtless see our own internal wranglings as opportunity....If I lived in Estonia, Latvia or Lithuania I would be very concerned. Cameron was ridiculed for the pointing it out, but a destabilised EU certainly makes war on continental Europe for the first time since the Balkans look like a much more serious risk, and small wars can sometimes turn into much bigger wars as we discovered last century.
Random things:
The EU has many different programmes from which we benefit in Arts/Culture, education, science/medicine and research. Unless the ultimate resolution is that we (like Norway) are not officially a member but continue to make all the payments and permit all free movement then you would have to expect that Erasmus programmes and the like will no longer be open to us and that many current research programmes will no longer be funded. I know of one research programme developing a cancer treatment protocol that expects to be shut down at the end of this year when the current phase ends, despite very promising results, only because someone I know was discussing it recently, but there will be countless others.
There will be some benefits - some of the EU agreements mean that we have suppressed some industry sectors as a trade off to benefit others, or in return for subsidies. We could (if we wanted) disregard things like agreed EU fishing quotas...someone more well up on it could probably list a ream of ways farming and manufacturing will benefit by being freed up, but the reality is that when we start to negotiate trade agreements we will have to make concessions again anyway.
So, right now nobody really knows anything for certain except the initial financial losses as they appear and as recessions are measured in quarters it could be the winter before we know if this initial uncertainty is going to do real damage. It could be a further two years before the negotiations are concluded and we know what the exit terms are and exit changes come fully into force. I imagine it will be 5 years before anyone can say exactly what the effects are, and a decade before we can fully evaluate the effects of the decision.
All of the above probably makes it clear that I voted to stay...a 'leave' voter may be able to provide some positive benefits that will offset some of the above. I'm in a very lucky position in that I have another EU citizenship and will relocate there. We were older parents, so our children are young enough for this crisis to have levelled out before they get to the career stage of their lives. I feel desperately sorry for the current crop of school leavers (and for the future generation of grandparents who may see their grandchildren raised on the other side of the world) and also for those who may find themselves trapped in negative equity if things get grim and recession takes hold.