At the fundamental level, what's happening in Italy is essentially what's been happening in Greece. Both countries have been running too high a deficit for too long. Since neither has power over the currency that they use (unlike the US and the UK) both countries are dependent on going to the markets to finance their debt.
The problem is that investors want to get paid back. If investors perceive a higher risk in the Italian bond market, then that makes the costs of the Italian debt go higher. Eventually, the costs of financing this debt become too much, and Italy must either cut back services & entitlements (this is called austerity) or default on it's debt obligations (called - wait for it - default).
Italy is another country in the PIIGS group - Portugal, Italy, Ireland, Greece, and Spain. These euro-zone countries are seen as the "most risky" in the zone, and that's what everyone is keeping an eye on. Most of the noise has been made in Greece, but with financial problems coming up in Italy, there's real concern about the overall stability of the Euro-zone.
One of the problems is that all of these countries are interconnected somewhat. So what happens in Greece affects what happens in Spain, which affects what happens in Italy, which affects what happens in France, the UK, and Germany. This is the "interconnectedness" of the system that I've posted about in the past.
The real concern is that a quick default by one of the PIIGS leads to another cascading series of problems in the financial sector, which would then either result in some serious problems in various financial companies around the world, or additional bailouts. That may strain, to the breaking point, the EU as a whole. Further, it may significantly damage the EU and world economy. Which is why everyone, including the US and the Chinese, are watching this intently.
Here's some good articles from the BBC if you want to read up further...