Firstly note that no computer system connected to another computer system
is "secure"
For security do not connect a computer to any network.
That said let's add to it.
This Apache webserver that you have asked this page from had a security exploit
that could cause the person exploiting it to do anything on the server including
change these pages. Whereas this has been resolved it's probably not the last or
only exploit out there.
There are ways of ensuring security - one way is by using encryption.
The strenth of Encryption depends on the algorithm used.

HTTPS servers encrypts that session between Client and Server, therefore by using
a 128bit key anyone intercepting your traffic will first have to decrypt it.
Some sites go a step further and encrypts the data received on an encrypted
connection :
eg. CHIP.CO.ZA and TAX.CO.ZA

When doing "secure" transactions such as banking etc. information exchanged
with an HTTPS server is encrypted.
This of course means nothing if the server itself is compromised.

Connecting a Computer to a public network like the Internet results in this computer
being visible to other computers on the same network. There is no way to be
"invisible" on the Internet. You can use a firewall that drops new connections or unknown packets
and do any number of smart things but on the network layer you will still be there...
I guess security is a huge field and impossible to cover in a paragraph. The best advice I
can give the average surfer out there is :

WINDOWS Users :

1. Run a firewall of some sort
2. Don't run any services (unless you know what you're doing
3. Disable Netbios over TCP ;)
4. Use an Antivirus program - and update it religiously
5. Be paranoid, very paranoid. If you suspect something it probably is so.
6. Make sure you aren't using an easily predictable TCP sequence (windows95/98 and others)
7. Ask a friendly Linux user to scan your box and tell you of weaknesses.
8. Be sure to update your Exploder and Outlook regularly from the Microsoft site. (or use Netscape)


LINUX Users :
1. Disable services you aren't using.
2. Don't laugh at the windows users.

Any other suggestions ?