“Permissions” are a slightly-exposed part of the underlying Unix operating system. They determine who can read, write, execute, or delete (these four are independent) a given file. They’re largely meaningless in the Mac world (although you can see them in the Get Info window), but there are lots and lots of system files that get modified constantly, and occasionally their permissions get messed up. You can repair them using Disk Utility. You don’t need to do this routinely; even if they’re messed up it’s usually no big deal. Repairing permissions is one of those time-killing steps Tech Support likes to have you do now and then while they figure out what the real problem is. (Kidding aside, this sometimes fixes issues, but not often). Repair them when you’re experiencing “weird” problems, or every couple of months if you like the peace of mind it brings you (I do it about once a year, just because I happen to be in Disk Utility and thing “gosh, haven’t clicked that button in a while.”)
There are effectively no Mac viruses “in the wild,” no matter what you’d think by reading the news. But you can pass on Windows viruses by mail – they won’t infect you, but they can move on to other systems. A little caution will solve this. Some Microsoft Office Macro viruses can also propagate via the Mac, but those aren’t very common any more. Don’t worry about sites you visit infecting you (it’s a pretty remote possibility), but DO worry about things like “phishing” sites that pretend to be something they’re not in order to elicit information from you. These usually come in the form of emails telling you that your bank needs you to update security information, and go to this web site…
The occasional terminating application isn’t cause for worry; Safari seems to be a common culprit here, in any case. It is possible to get the TCP/IP stack (the software that underlies your Mac’s connection to the Internet) confused, but a reboot will fix it right up. Internet access problems are only rarely the computer’s fault, though - especially when normally-working systems stop. They’re usually transient, and will go away in a few hours or a day or so.
Macs are not completely immune to malware, but they’re so close that unless you’re actively trying to get yourself compromised, it’s pretty tough.