I have a old computer (p4 with 128mb RAM) in my home and I have installed Red Hat Linux 9 in it. I have a broadband connection (BSNL DataOne) which requires a username and password to connect.

I am using this connection on Fedora 9 in my laptop and it works fine but I am unable to configure it here in Red Hat Linux 9.

I opened the Network Configuration and added a new xDSL connection. I typed the "Provider Name", "Login Name" and "Password" and then finished the wizard. It worked fine for Fedora but it is not working here in Red Hat linux 9. It just don't gets activated.
I have the Zebronix ethernet card which came with a installation disc but the driver is only 4 windows.My computer detected the ethernet card in Red Hat.

Given that RH9 (codename Shrike) hasn't been supported/updated in years, you might be better off getting an up to date distro. For small system like that maybe Damn Small Linux or Puppy.
Recent distros tend to have better HW support and be more debugged.
Also of course, RH9 will have security holes (lots).

Now, I am using puppy linux 4.1

I did downloaded today and installed in my old PC.

I am able to set up an internet connection (Main Purpose) but I am unable to browse the net using the browser or in any application.
the eth0 and ppp0 connections are blinking in the taskbar.

details of connection:
(I have a Broadband connection which requires a username and password to connect)

type: PPPOE
username: ...
Ethernet Interface: eth0
Activate on demand: No (the timeout sec is not working)
DNS address: Supplied by ISP's server
Firewalling: NONE

The connection is successful but I am not able to browse the net using the browser (SeaMonkey 1.1.11) or any other application.