By now it is likely that everyone has read about the crisis in Cyprus and the basic back story surrounding those events. So for the sake of brevity, I will skip the news aspects of the Cyprus crisis and skip right to the analysis.
Essentially, we can look at the Cypriot disaster using basic supply and demand. The graph below shows the pre-crisis state of the Cypriot financial system. Essentially, supply and demand for funds in Cyprus were in balance. Cyprus has a small domestic economy (about the size of the Scranton/Wilkes Barre, PA., area in the U.S.), and so it really only need a small domestic banking sector. However, by passing various regulations that eased taxes, legal scrutiny of funds, etc, the Cypriots were able to attract large amounts of foreign deposits to their banks. However, because they only had a small economy, they didn't realistically have enough productive uses of those funds in their own country. Instead, the banks in Cyprus, particularly the three largest banks, took to making property loans to residents in other European countries, especially Greece.
Now, all of that was OK as long as people continued to have confidence in the Cypriot banks and the loans continued to be paid by the Greeks, Italians, Spanish, and the Cypriots themselves. However, once the recession started, property values throughout Europe fell dramatically. This resulted in loans made by Cypriot banks starting to go bad. Eventually, enough loans went bad that the assets held by the Cypriot banks (the loans they made) were worth less than their debt outstanding (the depositors' money). Once this happened, the banks were essentially bankrupt, and the government of Cyprus was faced with two choices; shut down the banks, or apply to the EU for a bailout. Once people discovered the banks were in trouble though, depositors would rush to take their money out of the banks (resulting in the demand for funds shifting out), and depositors would no longer put new money into the banks (resulting in the supply of funds shifting back). As a result, the banks faced a shortage of funds - basically they don't have enough resources to pay back all of their depositors. That's why the banks were forced to shut down last week and why capital controls are very likely to be implemented when the banks reopen.
So that takes us to the stocks that will be most affected. To start with, because Cyprus is a small economy, it is likely that relatively few companies will be directly impacted in a meaningful way. However, where we could see greater impacts is in European market contagion. Specifically, banks with exposure to Europe are most likely to be affected. European banks like UBS (UBS), CSFB (CS), Banco Santander (SAN), Banco Bilbao (BBVA), Royal Bank of Scotland (RBS), Deutsche Bank (DB), and others could all see withdrawals if depositors decide that these banks are at any risk of failing. This is particularly likely in countries like Spain and Italy where the economy is weak, but it is still a (remote) possibility for the Swiss, German, and British banks. These risks hold regardless of whether or not Cyprus gets bailed out. If Cyprus gets bailed out, then it is probably less likely that bank depositors will become concerned and yank their funds, but it is still possible.
The big U.S. banks all have significant exposure in the area including Citi (C), JP Morgan (JPM), and Banc of America (BAC). However, it is less likely that these banks will face significant withdrawals since they are not connected to the EU. Indeed, what's entirely possible is that these banks could see significant inflows of cash. If Europeans decide that their own banks are at risk of failing, then they are likely to withdraw their funds and stash those funds in the U.S. banks (which are presumably safer given the U.S. economy is recovering). However, this by itself creates problems for the U.S. banks as they will then have to pay interest on these new deposits and they incur some back office costs associated with holding the deposits. Normally this wouldn't be a problem, except that many U.S. banks already have too much in the way of deposits and not enough good loan opportunities. As a result, if European depositors start removing their money from European banks, the U.S. banks are likely to take a hit to their profits regardless of any loss in value they might incur on their own investments in the EU.
Less directly, firms like Jeffries (JEF), AIG (AIG), Goldman Sachs (GS), Morgan Stanley (MS), and most other large financial firm could all be hit if withdrawals from European banks begin in earnest. All of these firms hold fairly substantial amounts of debt backed by European entities and any weakening of the already anemic European economy is sure to cause at least some trouble for the profitability of these firms.