Although it hasn't happened before, and it is very undesirable, it is possible for Bitcoin to split into two non-negligible separate currencies.
Why is this possible? / What causes a split?
There is no actual BTC in your Bitcoin wallet: there are only private keys which are used to unlock and then transfer ownership of BTC stored in the Bitcoin system. BTC never actually leaves the Bitcoin system. A split creates a clone of the original cryptocurrency, but with modifications preventing the two cryptocurrencies from actually being the same; after the split, you cannot send coins from one side of the split to the other. After the split, your private key can be used to unlock your coins on both the original system and the modified system.
It's like if someone got their hands on a copy of the google.com database and created super-google.com starting with the google.com database but with different features. You could use the same login credentials to login at either google.com or super-google.com. But after the clone was created, if you received mail.google.com email, it would not show up at mail.super-google.com, and vice-versa. Similarly, you can use the same private key to spend money on the original currency or the modified currency, but after the split they diverge.
Anyone can create a split. It only requires a few lines of code changes. But splits only matter if people of economic significance actually use the split (ie. they must run the modified software).
Splits happen in the following situations:
What happens to my bitcoins in a split?
It's similar to a stock spin-off. You have x BTC beforehand, and afterward you have both x BTC and x "BTC-X". But very importantly, your wallet will not magically know that it is now able to unlock both currencies. If your wallet is not updated to account for the split, then you will only be able to spend one of the currencies, probably the more "status-quo" one. Another possibility is that your wallet could be updated, but only to support a different currency than it would've otherwise, not to support both. It is not unreasonable for wallets to support only one currency, since they really weren't originally designed/intended to support multiple. But if your wallet only supports one of the currencies, then you will usually end up throwing away some or all of the other currency when you next send coins after the split, as a side-effect of the transaction -- this is called "replay".
Value is unpredictable. Like a stock split or spin-off, you'd vaguely expect the value of BTC immediately before the split to equal the sum of the values of "BTC" and "BTC-X" immediately after the split. But if a split like this is anything close to a 50-50 split (in relation to economic adoption/value, not node or miner adoption), then the whole thing will probably be unbelievably, catastrophically messy, which may result in the combined value crashing. On the other hand, if the minority currency is pretty small and things don't get too messy, the minority side could be speculatively overvalued (similar to altcoins) without affecting the majority side too much, causing the combined value to rise somewhat.
Which side of the split is the real Bitcoin?
That depends on a wide variety of factors, and to some degree it is subjective. The most important factor is which currency people/businesses/exchanges accept: if for example one currency is accepted by 95% of the pre-split economy, and the other is accepted by only 5% of the pre-split economy, then the 95% one is probably truly Bitcoin, and the other one can be considered a Bitcoin-derived altcoin. Note that mining power has very little influence here.
Some might say that both currencies post-split are valid incarnations of Bitcoin, but it is my philosophy that only one Bitcoin can exist at any one time; any uncertainty is only a temporary feature, and at least in hindsight there will be a single unbroken path from Satoshi's original Bitcoin to the current one.
What should I do to secure my bitcoins?
First of all, remove all bitcoins possible from banks/exchanges/"hosted wallets". If you can't export your private keys, then you don't actually control the bitcoins. This is a good idea in general, but it's especially important in case of splits. If you're using a Bitcoin bank, then you will have no control over what happens to your coins. Quite possibly, the bank will end up stealing/losing one of the currencies that you should have access to, either through greed/malice or technical incompetence, and the currency they throw away might well be the only one that ends up having any long-term value.
Possible splits are usually predictable some time in advance (see the next section). If in doubt, avoid sending transactions or trusting received transactions 12 hours before and up to a few days after the split time, and check the forum for more news. If a major split happens, I will very likely make a post and news item explaining how to handle the situation. Coins at rest are not at risk.
Generally, after a split:
What are SegWit, BIP148, UASFs, and "UAHF", and how do they relate to this? / When might a split happen?
SegWit is a set of changes to Bitcoin which increases the max block size, among other improvements. Under its initial BIP141/BIP9 deployment plan, there is almost no chance of a split.
A user-activated softfork (UASF) is a general method of deploying softforks (like SegWit). UASFs have been done in the past, both by Satoshi and after Satoshi left. Unlike the BIP9 deployment method originally planned for SegWit, a UASF does not require any miner cooperation. A UASF can fail, but only through insufficient economic adoption, not by any miner action. However, splits are more likely with UASFs than with BIP9 deployments, especially when the UASF is done on a compressed schedule. (For ideally-low split risk, a UASF would take about a year to activate.)
BIP148 is one specific UASF. It is intended to activate SegWit. It activates and may cause a split on August 1. If it succeeds economically, it will activate SegWit by November.
The large Bitcoin miner Bitmain announced that they might engage in a (nonsensically-named) "user activated hardfork" ("UAHF") shortly after BIP148 activates if BIP148 has any success. This would definitely cause an additional split, resulting in up to three currencies: BIP148-Bitcoin, Bitmain-coin, and (depending on the degree of BIP148's success) status-quo-Bitcoin. As mentioned previously, if you have x BTC before the split, you'll be technically able to claim x of all three of these currencies after the split.
The next time when a split is predicted to possibly happen is therefore August 1, 2017 at midnight UTC.
OK wallets
The following wallets give you some theoretical way of exporting your private keys, so they should be safe to hold BTC during Aug 1. Depending on what happens after Aug 1, you may have to export your private keys and use different software, though. (Do not go doing this willy-nilly in a panic, though -- wait for agreement and clear instructions from several sources. There will be no rush.)
The ones in bold are ones that I also tend to like for other reasons. The ones in italic seem to make it difficult to export your private keys, or I particularly dislike them for other reasons.
Mobile
Electrum
Mycelium
Copay
Bitcoin Wallet for Android
breadwallet
Simple Bitcoin Wallet
Bither
ArcBit
Coin.Space
Desktop
Bitcoin Core
Electrum
Bitcoin Knots
mSIGNA
Armory
Bither
ArcBit
Copay
Hardware
Ledger
Trezor
Digital Bitbox
KeepKey
Web
Green Address*
Coin.Space
