LFA - Loop-Free Alternate(s) (also sometimes called LFA-FRR which may add to your confusion) is a feature of IGPs such OSPF and IS-IS to allow nodes to calculate alternative paths to each prefix in the event of a link failure. These backup paths are maintained by each node and rapidly installed upon link/node failure so that traffic flow is preserved while the SPF algorithm is re-run.
In the case of an MPLS network, LFA ensures that the IP underlay of the network recovers quickly, preventing, say an LDP-based path from re-converging until the underlay is again stable.
FRR - Fast Re-Route is the name traditionally given to a similar process used in RSVP-based MPLS networks. Headend nodes for an LSP configured with FRR calculate an alternate, diverse (where possible) path for the LSP to follow, and quickly switch to it in the event of the primary path failing.
To summarise though, the difference between LFA-FRR and MPLS-FRR is essentially that LFA-FRR operates on the underlying IP network, while MPLS-FRR operates on the Label-Switched overlay (provided you are using RSVP for label distribution).