Important Note: This Answer Is Very Out of Date
A network of clients is organized as described in the foregoing FAQ. Threshold Relay produces an endogenous random beacon, and each new value defines random group(s) of clients that may independently try and form into a "threshold group". The composition of each group is entirely random such that they can intersect and clients can be presented in multiple groups. In DFINITY, each group is comprised of 400 members. When a group is defined, the members attempt to set up a BLS threshold signature system using a distributed key generation protocol. If they are successful within some fixed number of blocks, they then register the public key ("identity") created for their group on the global blockchain using a special transaction, such that it will become part of the set of active groups in a following "epoch". The network begins at "genesis" with some number of predefined groups, one of which is nominated to create a signature on some default value. Such signatures are random values - if they were not then the group's signatures on messages would be predictable and the threshold signature system insecure - and each random value produced thus is used to select a random successor group. This next group then signs the previous random value to produce a new random value and select another group, relaying between groups ad infinitum and producing a sequence of random values.
In a cryptographic threshold signature system a group can produce a signature on a message upon the cooperation of some minimum threshold of its members, which is set to 51% in the DFINITY network. To produce the threshold signature, group members sign the message individually (here the preceding group's threshold signature) creating individual "signature shares" that are then broadcast to other group members. The group threshold signature can be constructed upon combination of a sufficient threshold of signature shares. So for example, if the group size is 400, if the threshold is set at 201 any client that collects that many shares will be able to construct the group's signature on the message. Each signature share can be validated by other group members, and the single group threshold signature produced by combining them can be validated by any client using the group's public key. The magic of the BLS scheme is that it is "unique and deterministic" meaning that from whatever subset of group members the required number of signature shares are collected, the single threshold signature created is always the same and only a single correct value is possible.
Consequently, the sequence of random values produced is entirely deterministic and unmanipulable, and signatures generated by relaying between groups produces a Verifiable Random Function, or VRF. Although the sequence of random values is pre-determined given some set of participating groups, each new random value can only be produced upon the minimal agreement of a threshold of the current group. Conversely, in order for relaying to stall because a random number was not produced, the number of correct processes must be below the threshold. Thresholds are configured so that this is extremely unlikely. For example, if the group size is set to 400, and the threshold is 201, 200 or more of the processes must become faulty to prevent production. If there are 10,000 processes in the network, of which 3,000 are faulty, the probability this will occur is less than 10e-17.
As well as being incredibly robust, such systems are also highly efficient. In a broadcast gossip network, a group of 400 can produce its threshold signature by relaying only about 20KB of communications data. Meanwhile the BLS threshold cryptography libraries DFINITY was involved in creating can perform the computation for the necessary operations in fractions of a millisecond on modern hardware. You can learn more about Threshold Relay by reading the DFINITY Consensus Whitepaper.