PCI is Parallel. There are several versions including a PCI-X standard that was quite short lived. They are not compatible with PCIe
PCIe (PCI Express) is a serial connection. More pins means more bandwidth, and each generation basically doubles the communication speed.
PCIe 2.0 x16 slot = PCIe 4.0 4x.
So those 1x slots at PCIe 3.0 are still quite fast, and why they haven't really needed to get bigger. That does mean when you start looking at more recent PCIe cards it can be worthwhile to check the PCIe revision.
On a more recent motherboard you will find the following:
PCIe x16 - main slot and secondary x16 slot generally connect directly to the CPU.
PCIe x16 slots (8x connection), Typically this is shared with the first x16 slot, and each slot will run at 8x when ANY card is plugged into the second slot.
PCIe x16 slots (4x connection). These are usually connected to the motherboard chipset, not the CPU. Just gives you an option for larger cards if needed.
PCIe 1x slots. Pretty much always connected to the chipset, it is common for the x16 (4x connection) slot to share connections with the other 1x slots and potentially SATA ports on the motherboard.
M.2 slots. 2280 and up are generally for mSATA and NVMe SSDs. These can be connected to the CPU or chipset. These can share connections with SATA ports, GPUs, and all sorts of things. Depends on the motherboard.
Shorter 2242 M.2 slots are usually wired for 1x and are for WiFi modules and connect to the chipset.
Internally, all other devices are generally connected to PCIe lanes on the chipset. Internal network adapters, USB controllers, SATA controllers, audio chipsets, etc.
That little extra bit on that WiFi is just a guide so that you don't plug the card into the middle of a slot or something else odd. The card CAN go into an x16 slot, but you don't have to. If you have an open 1x slot, that would make more sense, if you needed it anyway.
If you are having trouble installing a card, it is likely you might have one of the types of cases where the metal bracket to screw it down has a guard/shield and/or is actually outside the case.