An interface is a connection to a local segment. Generally, it's a network interface or port (layer 1 and 2). A single network port may have multiple IP addresses and represent multiple layer-3 interfaces though. Using VLAN tags (802.1Q), it may also represent multiple layer-2 interfaces.
A gateway is a potential path to a remote network. It needs to be in reach of one or more interfaces, ie. be part of the same segment and subnet. The default gateway is the path to 'everything we don't have a better path for'.
Gateway is the functional name from the perspective of a local segment where a gateway leads 'elsewhere'. Router is the very same thing from the overall perspective of the larger network - a connection between two or more subnets.
Now, the interface in a routing table is that interface which is to be used to talk to the given gateway. A host may have multiple interfaces with the connecting segment and normally chooses the one with the highest link speed (lowest metric).