Hypothetically we have a small network with a router, a switch and a network cards. All of them are 1GB, but one. Is this device (i.e. network card) going to downgrade my 1GB network, or only the last connection between the network card and the last switch would have an effect on?

If you only have a card with a 10/100Mbps connection and the rest are 1GB connections, then unless you're using the slower card as a network server or are generally sending lots of data to and from it, it shouldn't make any difference to the rest of the network, said card will just run at the slower speed.