You might find various ones second hand. For the most part it is not a bad thing for DS flash carts like it might be with a lot of other things -- for things in the DS slot there are rarely any batteries or things that wear out that can't just be cleaned and returned to use.
1) Supercard kind of stopped doing anything many years ago now. This is one reason why the DSTwo is so expensive (the other being its onboard processor that allows extra homebrew). It even meant their normal flash cart (the Supercard DS One) became hard to find. However more recently clones of them appeared
https://gbatemp.net/threads/dsone-clones-appear-to-be-available-for-purchase.586226/ so there is that option.
2) Acekard made flash carts, however they released the code to allow others to do things for a bit so there ended up a user made kernel for it known as AKAIO. AKAIO had the best compatibility, the fastest and all manner of other nice things. To that end the Acekard 2i, also known as the AK2i, became one of the best flash carts to have. I have not checked in a while but I did see some new in shops in fairly recent times.
3) R4 was a flash cart early in the DS slot (as opposed to GBA slot) flash cart scene. It seemed to have the right blend of price, ease of use and capabilities that it became super super popular. So popular that "R4" became the term many referred to flash carts as (similar to how jailbreak these days means any hack as far as some people are concerned rather than a specific thing for Apple products). These flash carts with R4 in the name are known as R4 clones despite some not actually being clones of the R4 (you can actually get copies of the original R4 but don't). In more recent times it was seen to be popular to have the device read the system date and if it is beyond a certain date stop working, something known as timebomb. You can work around timebomb by setting the date on your DS or using another kernel that does not do that but easier to find one without the thing in the first place. Preferably one that also supports the Wood firmware, which is based on AKAIO discussed above and thus quite good.
4) There was way more than supercard, acekard and R4 doing DS flash carts. EZFlash (their last one being the EZ5i), M3 (whose thing was the basis for the R4 actually) with the M3i, ismart, team cyclops and many many more (
http://linfoxdomain.com/nintendo/ds?mode=stats is not complete but still good).
Many of these stopped being updated before the last DS ROMs got released so might not play every commercial game without further fiddling where the ones in 1), 2) and 3) above will.
https://gbatemp.net/threads/ap-patch-preservation.477536/ can get things working. The usual cut off point where things stopped getting updates was also when they were blocked on DSi and 3ds, about the time pokemon conquest was released if you want to look at release lists
http://advanscene.com/html/index.php
To that end if you found one of these later cards on a facebook group, vendor somewhere or whatever you could still run homebrew, most games, probably have some cheats and other nice features (the EZ5i for instance being one of the few things to compete with Supercard and even beat it in some cases when it comes to extra features).
Enhanced flash cart is a term the community uses to refer to things like the DSTwo. These are DS flash carts that rather than just play games actually had processors of their own onboard that homebrew could use to get extra power and do more than a normal DS. The DSTwo is the only example that really matters but there were two others that technically could do more, the ismart MM/ISMM with its port of Dingux (a version of Linux that has emulators built in, including some arcade stuff) being that.
I see now I have just done more words but hopefully it clears some things up. About the only other thing I would note is GBA slot stuff. There are some emulators that try to do GBA on DS slot flash carts (the DS slot is way too slow to do it natively) but they are only ever likely to be proof of concept rather than something you would want to use to play games. To that end if you want to run GBA ROMs and homebrew (and there are a lot of obscure emulators for the GBA that never made it to the DS) you will want a GBA slot flash cart. These days that mostly means find a EZFlash 3 in 1 and probably replace the battery (also mod the case if you want it to work on an original DS), an EZ4, EZ Omega, EZ Omega DE, or Everdrive GBA with the latter three being quite expensive (way more than the DSTwo that kicked this whole conversation off) but also very good.
