A NVMe, AHCI and IDE are transfer protocols (languages). They run on top of transfer interfaces such as PCIe or SATA (spoken, written).
SATA is the market incumbent and dominant interface for connecting an SSD to the PC. It employs the command protocol AHCI (it also supports IDE) which was built with slower spinning disks in mind rather than flash memory. SATA transfer rates begin at 150 MB/s and max out at 600 MB/s for third generation technology. For most consumers this is still absolutely adequate.

PCIe (PCI Express) supersedes SATA as the latest high bandwidth interface. Entry level PCIe SSD speeds are two to three times faster than the older generation of SATA 3.0 SSDs mainly due to the number of channels contained by each to transfer data (roughly 10 for SATA and 25 for PCIe). However, depending on usage, real world benchmarks may not reflect this massive gain due to bottlenecks elsewhere in the PC. Here is a comparison between the two market leading NVMe and SATA SSDs: Samsung 970 Pro vs 860 Pro.

NVMe is the latest high performance and optimized protocol which supersedes AHCI and compliments PCIe technology. It offers an optimised command and completion path for use with NVMe based storage. It was developed by a consortium of manufacturers specifically for SSDs to overcome the speed bottleneck imposed by the older SATA connection. It is akin to a more efficient language between storage device and PC: one message needs to be sent for a 4GB transfer instead of two, NVMe can handle 65,000 queues of data each with 65,000 commands, instead of one queue that with the capacity for 32 commands, and it only has seven major commands (read, write, flush etc). As well as delivering better throughput NVMe offers reduced latency. NVMe will be the protocol of choice for the next generation of storage technologies such as 3D XPoint.

The Best.
CPUGPUSSD
Intel Core i7-8700K $320Nvidia GTX 1070 $413Samsung 850 Evo 250GB $104
AMD Ryzen 5 1600 $175Nvidia GTX 1060-6GB $250Samsung 850 Evo 500GB $138
Intel Core i5-8600K $222AMD RX 480 $280Samsung 850 Pro 256GB $125
HDDRAMUSB
Seagate Barracuda 1TB (2016) $44Corsair Vengeance LPX DDR4 3000 C15 2x8GB $177SanDisk Extreme 64GB $44
Seagate Barracuda 3TB (2016) $80G.SKILL Trident Z DDR4 3200 C14 4x16GB $911SanDisk Extreme 32GB $30
Seagate Barracuda 2TB (2016) $59HyperX Fury DDR4 2133 C14 2x8GB $226SanDisk Ultra Fit 32GB $12
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