Redundancy .... 1+1, 1:1, N+1, N:1, N:N ???

1:1 & 1+1

1+1 redundancy typically offers the advantage of additional failover transparency in the event of component failure. The level of resilience is referred to as active/active or hot as backup components actively participate with the system during normal operation. Failover is generally transparent (disruption to system availability) as failover does not actually occur (just degradation to system resilience) as the backup components were already active within the system.

Examples of 1+1 redundancy:

• Dual active power supplies in a server.
• Mirrored hard drives within a server/PC system.

Both 1:1 and 1+1 schemes have one active device or line protected by a redundant device or line. The difference between the two schemes is whether or not reversion takes place after the fault is cleared. 1+1 means that every component has one dedicated backup. Each component can be replaced by the one and only other backup device. Minimum number of components is 1.

For example, device B is protecting device A in a 1:1 configuration. 
If device A fails, device B will become the active unit in the configuration. 
When the fault at A is cleared, device A once again becomes active after a predetermined time and B returns back to standby (idle) mode. Reversion takes place from B to A. 

For example; device B is protecting A in a 1+1 configuration. 
If device A fails, device B will become the active unit in the configuration. 
When the fault at A is cleared, device B remains as the active device indefinitely (unless a fault occurs at B). No reversion takes place. 

The key advantage of 1+1 in telecom systems, if that only one disruption of traffic occurs. In 1:1 configurations, two disruptions take place; one when the fault occurs and a second when the fault is cleared and reversion takes place from the standby unit.

N:1, N+1 & N+N

N+1 redundancy is a form of resilience that ensures system availability in the event of component failure. Components (N) have at least one independent backup component (+1). The level of resilience is referred to as active/passive or standby as backup components do not actively participate within the system during normal operation. The level of transparency (disruption to system availability) during failover is dependent on a specific solution, though degradation to system resilience will occur during failover. It is also possible to have N+1 redundancy with active-active components, in such cases the backup component will remain active in the operation even if all other components are fully functional, however the system will be able to perform in the event that one component is faulted and recover from a single component failure. An active-active approach is considered superior in terms of performance and resiliency.

Examples of N+1 redundancy:

• Connecting devices (server etc) in dual switch SAN fabrics employ a discrete path to each switch. Only one path is active at any given time, resiliency is provided by the availability of an additional path if the active path becomes unavailable.
• Data centre power generators that activate when the normal power source is unavailable.

1+1 and 1:1 means the same, as well as N+1 and N:1. In general it is describing how many failures we can tolerate. 

N+1 means we have 1 backup device per group, so we can tolerate one failure. Any component can replace any of the components, but only once. Minimum number of components is N.
N+N means N backup devices per N components, and we can tolerate N failures. Any can replace any, minimum number of components is N.

One important difference between N+1 and 1+1 is that in a 1+1 system the standby server has the ability to maintain the state, e.g. in a phone network the standby server would be able to keep ongoing calls as it constantly monitors the state of all activities in the active server. In an N+1 system there's only one standby for N active servers and the call state is lost in case of a switchover. 
The terms 'hot standby' and 'warm standby' describes the same concept.