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That would either be a Crystal chip or a Matrix Infinity chip. They are both similar in functionality and cost about the same too.
You can play backups of both PS2 and PS1 games with both these chips and even play NTSC or PAL games no matter what console / TV you have.
You can load all kinds of different ELFs (runnable apps), like a media player (check PS2Reality Player or SMS), FTP Server or even arcade emulators, just to mention a few.
Older consoles (the big/fat models) do not have an integrated network card and will be harder to get new. You can install an internal HDD to them though.
(There are only a handful of games that support HDDs). Some people like to install games on the HDDs to avoid "wasting" the laser but I personally think it's not necessary if you buy a brand-new console (unless you're a 24/7 player).
The newer consoles (so called slim versions) include LAN and are 70% smaller.
There are known laser problems with some older models (like v12) so try to get at least a v13.
As a side comment I would advise you to keep your hands off of used consoles, specially those sold on ebay. The reason for this is that the PS2s laser on used units will be worn and probably have a re-adjusted voltage.
An original replacement laser will cost about the same as an old ps2 so it's not worth it.
I personally have a Crystal Chip and it works like a charm. One big advantage of this model is that you can play online games too.
Having a modchip means that you don't have to use the "swap trick" to get stuff to work. You can either build a CD from where you will load apps or copy and load them from your MC. The choice is up to you.
Finally I wouldn't recommend trying to install the chip yourself even if you have soldering skills. It's not worth the "experience" and you will probably just end up with a dead unit. If you're new to this whole PS2 scene and you get stuck ie. installing apps you'll never really know if it's because of a faulty self-installation or not. (Just read some posts here under "hardware help" to see what I mean). You have been warned!
Many stores will mod the console for you for a fairly low price and even offer you a warranty. Do some research and try to find a reliable mod-store.
Thank you very much I know someone who has installed lots of chips so I'll get him to do it. I'll get a Crystal Chip seeing as it works online
Just one last thing, do I need a specific model of PS2 as I heard some of them were very hard to mod and something about a laser fix?
Hi, I edited my post and extended it. Check it out.
About the laser fix: that is just a way to protect the laser on some older (slim) consoles. All slim versions from v13 up don't need it.
Thank you very much