"AHCI in BIOS" is a setting you can make - and SHOULD make sure it's correct.
History: the new SATA drive systems and Windows XP were released at almost the same time. At that time, XP was released fully able to deal with IDE devices through its "built-in" device driver (like previous Windows versions), and this was never changed in XP. It only was changed in Vista and Win 7, which added "built-in" drivers for the new drive type.
"SATA drives" refers mainly to the hardware system design. From the software side of using them, they are best treated as "AHCI devices". In addition to some hardware improvements that affect performance, AHCI devices also have some extra features that IDE devices never had. For any OS to use any device type, it needs a software driver that manages communication with them. Win XP, like previous Windows, has several "built-in" drivers for common device types like IDE drives, ATAPI optical drives, floppy drives. It does NOT have drivers for other types, such as AHCI devices, SCSI devices, or RAID arrays. It can have those device drivers installed into it once it is running, but that imposes a limit. To be able to use such devices, XP has to boot from a device (typically a HDD), get itself running, then read the required device drivers from that HDD and add them in. THEN it can start to use them. But that means it cannot BOOT from such a device.
For a long time, Windows has had a process to deal with this. When you first install XP, for example, one option presented early is to press the F6 key if you wish to add in some device driver(s) not already part of XP, and make them a permanent "built-in" driver for all future use. When you do this, that particular installation of XP on your machine now DOES have these extra device drivers "built in" and it CAN load them early enough to actually boot from a device of that type. In this way you can enable your Win XP to boot from a SCSI drive, a RAID array, or an AHCI drive. BUT the only way Win XP knows how to do this is from a FLOPPY drive containing a disk with the drivers on it already. So, to do this, you have to have a floppy drive as part of your hardware (at least temporarily when you first install Win XP) and a diskette on which you have previously loaded the driver file(s). Now, as it happens, about the same time as XP and SATA were coming in, floppy drives were disappearing and many people did not bother to install one in their new machines.
To solve that problem, BIOS makers included a "work-around". Very near the place in BIOS Setup where you Enable the SATA ports, there is a line for something like "SATA Port Mode", offering choices like "IDE (or PATA) Emulation", Native SATA mode", "AHCI Mode", and "RAID Mode". When using SATA drives, the ideal setting is AHCI mode so that all of the features and performance of these units can be used. If you are installing Vista or Win 7, this is definitely the way to set this option, because those newer OS's have "built-in" drivers for AHCI devices and can handle them just fine. But to use that in Win XP you must do the driver addition from a floppy drive, as above, using the F6 process. The alternative is to set this option to "IDE (or PATA) Emulation". In this mode, the BIOS intervenes and makes the actual SATA device behave like an older IDE device, and Win XP is perfectly happy to deal with that without any added driver. However, in doing this you lose a few of the advantages of AHCI devices, which may or may not matter to you.
In your case, I suspect you are using Win 7 as your OS, and not XP. So you should set the BIOS Setup option for SATA Port Mode to AHCI to take full advantage of the new design features.
I still have some reservations about this, though, and this has nothing to do with SATA Port Mode. You are using an older IDE drive via an IDE-to-SATA adapter on a SATA port. You say your boot times are several minutes long. Although a new SATA HDD will surely be faster than the old IDE drive, that would NOT explain a difference of 4 minutes versus 1 minute for a boot. There is something else taking a long time in your boot process. I know my machine is slow to boot, and I suspect that is because my grandson and daughter have loaded many games on it that check with their websites as the machine boots. I'm also suspicious that the automatic re-indexing of the hard drive files takes time. I expect to "fix" this when I upgrade to a new OS soon. But I'm sure I could not fix this with a simple HDD upgrade.