I'm currently on using a i3-4160 CPU and the total number of PCI-E lanes it supports is 16.
In addition to the 16 PCIe lanes from the CPU, there is also an interface called "DMI", this connects the CPU to the PCH, the DMI interface on your generation of CPU is basically equivalent to PCI express 2.0 x4. The PCH in turn provides a number of PCI express lanes for slower prehiperals. If all of these lanes were maxed out at the same time then the DMI interface to the CPU would be a bottleneck, but in practice this is not typically a huge problem. Only a couple of devices are typically maxed out at any given time.
An x1 slot is almost certain to be connected to the PCH, so it will not have any impact on the graphics card, though in some cases it may not be able to be used at the same time as some onboard features.
An x8 (electrical) slot on the other hand is most-likely to be connected via signal-switches to the lanes from the CPU. So putting a card in it will reduce the number of PCIe lanes available to the graphics card from 16 to 8.
An x4 (electrical) slot may be connected either way, you would have to study the motherboard's documentation to find out.
On some high end boards you may also find PCI express bridge chips. Unlike the crude signal switches mentioned above these allow the bandwidth to be dynamically shared between different devices.