The PS2 mechacon detects the copy protection found on original PS1 and PS2 CDs and PS2 DVDs, and will not allow discs from out-of-region to be read, nor will it allow access to burnt media (with the exception of DVD-Video and CD-Audio discs, the former of which is a small loophole which I'll explain later). The swap kits work because the mechacon only checks the copy protection when a disc is first inserted. On fat PS2s, slide cards are commonly used to force the drive tray to open without tripping the sensors that tell the mechacon that the drive has been opened. On slim PS2s, there are small pieces of metal that can be strategically placed underneath the lid of the drive, serving the same function (the system can also be disassembled and the sensors blocked that way). Alternately, for all types of PS2s, there are aftermarket "flip-top" lids that allow the drive to be opened without tripping the sensors. This is what allows Swap Magic to work - you boot with the desired type of disc (CD or DVD) and then after the SM screen comes up, you can use the above methods to swap out for a copied disc which will generally then boot fine. Swap Magic does not work with DVD-9 games, however.
Remember how I said that the mechacon allows access to DVD-Video discs? Well, that little factoid along with some handy homebrew tools will allow booting copied DVD games on an unchipped PS2. There's a little PS2 program called ESR that will boot such discs (the ISOs do have to be patched with a PC-side program that places a fake DVD-Video filesystem on the disc, which the mechacon sees and allows the disc to be read). The hardest part for some people is getting the homebrew to run. On the older fat PS2s, there is the "independence exploit", which works via a hacked PS1 game save and booting with said PS1 game in the drive. I've never used this exploit because there are better ones out there and the hardest part of it is getting the hacked PS1 save installed on the memory card (MC from here on out). There is also "Free McBoot" which is basically a PS2 softmod that works on all systems up to SCPH-9000x with date code of 8B (a few 8C systems support it, but it was during that quarter when Sony updated the PS2 BIOS to v2.30, removing the MC update check that allows FMCB to boot). Once again, the catch for many is getting the FMCB installer to run. My preferred method is to use a Swap Magic disc, as it will also boot automatically from a USB drive if a PS2 .ELF is named SWAPMAGIC.ELF and placed in a SWAPMAGIC folder. This can be anything but I'd recommend a file browser such as uLaunchELF, which then would allow you to browse the USB drive and run any other homebrew. This will work even on BIOS v2.30 machines where FMCB won't boot (if one tries to install it to the MC, it will successfully install, and in fact that MC can then be used to boot FMCB on any pre-v2.30 system, but will not boot on the system it was installed on). FMCB must be "installed" to MCs, it cannot be copied from an installed card to another card, as the installation signs it to the specific MC via MagicGate (for this reason I only recommend Sony official 8MB cards for use with FMCB, or newer non-bootdisk Datel Max Memory cards).
Owners of fat PS2s with network adapters have the greatest advantage of all, however (so much so that I, already owning a slim PS2, purchased a used fat PS2, network adapter, and 250GB IDE hard drive). With FMCB installed and a little piece of homebrew called Open PS2 Loader (OPL), for most games there is no need to even use the optical drive at all, as OPL allows loading of games from three sources - USB drive (slowest and least compatibility but also works on slims), Windows-style SMB/CIFS file share (medium speed, better compatibility, works on slims), or internal HDD (fats only, best compatibility, fastest load speed). OPL also allows the use of virtual MC files stored on the same medium as the game is using. There's also another piece of homebrew called GS Mode Selector (GSM) that allows forcing the PS2 into any of several video modes, including 480p, 576p (on slims only with the current version), 720p (admittedly due to either hardware limitations or lack of knowledge this mode is pretty much useless), 1080i, and even some VGA modes with an appropriate cable. These modes of course require a component cable, they do not increase the actual game resolution (except for certain games that don't have a 480p setting but that still render internally at 480p, running them in 480p can increase quality) but they do help reduce lag on HDTVs and make the picture sharper as the set is doing less scaling. GSM doesn't work with every game (in fact, some games will just lock up with GSM is forcing one of the HD modes) but on games that it works with, the 1080i mode is pretty awesome (although on most TVs it will also result in "forced widescreen", so if the game doesn't support widescreen then it will be stretched). If you do get a fat PS2/network adapter/HDD combination, avoid Western Digital drives, as the Molex/IDE header spacing is incompatible with the network adapter (the NA itself can be modded to slide the Molex connector to make it fit but I prefer using drives that fit by default). I have successfully used Maxtor and Samsung drives with no issues.
All of this can be done even on BIOS v2.30 slims as I mentioned, but it would require the Swap Magic disc to always be used as a boot disc, whereas the older systems only require an FMCB-installed MC and no use of the optical drive. I haven't used my fat PS2s optical drive since I bought it (as I already had an FMCB-installed MC from my slim). Typical of Sony to take away methods of prolonging our systems' longevity (because on BIOS v2.30 systems, once the optical drive dies as it is eventually liable to do, the system would require a physical modchip installation to be able to boot homebrew).
Lastly, I will make a public statement. Don't pirate games. I realize saying this here doesn't make much difference (if one's gonna do it on their own, I nor the mods can stop it). However, this would still protect Retro and myself from any liability, however slim the chance. I do make an exception for prototype builds of games - if they're not bit-identical to the retail equivalent, then download away for preservation and documentation purposes. ImgBurn is good for ripping DVD games for either ESR or OPL usage, while ESR can not boot CD games (as the exploit depends on the fake DVD-Video filesystem which the mechacon won't fall for on a CD). ImgBurn is good for ripping PS2 CD games to ISO format (instead of the common BIN/CUE which may be necessary for some games to boot using the Swap Magic method which will allow copied CD games).
tl/dr; PS2s can boot burned games without a modchip. However, there are limitations depending on which method is used.