RSRP is the linear average of reference singal power (in Watts) accorss the specified bandwidth (in number of REs). This is the most important item UE has to measure for cell selection, reselection and handover. You can think of this as the one similar to CPICH RSCP in WCDMA.
The formal definition of RSRP in 3GPP 36.214 is as follows :
Reference signal received power (RSRP), is defined as the linear average over the power contributions (in [W]) of the resource elements that carry cell-specific reference signals within the considered measurement frequency bandwidth. For RSRP determination the cell-specific reference signals R0 according TS 36.211 shall be used. If the UE can reliably detect that R1 is available it may use R1 in addition to R0 to determine RSRP.
Note : R0 is the Cell Specific Reference Signal for Antenna Port 0 and R1 is the Cell Specific Reference signal for Antenna Port 1 (Refer to 36.211 Figure 188.8.131.52-1. Mapping of downlink reference signals (normal cyclic prefix))
Following is an example of one downlink radio frame. The red part is the resource elements in which reference signal is being transmitted. RSRP is the linear average of all the red part power.
Since this measures only the reference power, we can say this is the strength of the wanted signal. But it does not gives any information about signal quality. RSRP gives us the signal strenth of the desired signal, not the quality of the signal. For quality of the signal information another parameter called 'RSSQ' is used in some case.
UE usually measures RSRP or RSRQ based on the direction (RRC message) from the network and report the value. When it report this value, it does use the real RSRP value. It sends a non-negative value ranging from 0 to 97 and each of these values are mapped to a specific range of real RSRP value as shown in the following table from 36.133.
For practical calculation example, refer to RSRQ Calculator (YouTube)
Just to relieve a little of headache to figure out the mapping between the reported RSRP and measured value in dBm, I just made precalculated table as shown below.
 3GPP - 36.214 Evolved Universal Terrestrial Radio Access (E-UTRA); Physical layer - Measurements
 3GPP - 36.133 Evolved Universal Terrestrial Radio Access (E-UTRA);Requirements for support of radio resource management