Use of 8AWG Cu in 1"ID grey PVC buried for 120' on 50 amp circuit
Note: All block quotes are sections from NFPA-70:2014 (aka 2014 NEC)
1st problem: 8 awg NM can only carry a max of 40 amps
NM is treated as a cable assembly with 60°C conductors per 334.80 even though the individual conductors are typically rated at 90°C:

334.80 Ampacity. The ampacity of Types NM, NMC, and NMS cable shall be determined in accordance with 310.15. The allowable ampacity
shall not exceed that of a 60°C (140°F) rated conductor. The 90°C
(194°F) rating shall be permitted to be used for ampacity adjustment
and correction calculations, provided the final derated ampacity does
not exceed that of a 60°C (140°F) rated conductor. The ampacity of
Types NM, NMC, and NMS cable installed in cable tray shall be
determined in accordance with
392.80(A).

This means that the max 8 awg NM cable can carry is 40 amps per table 310.15(B)(16)




2nd problem: NM cannot be used underground
The underground conduit is a wet location per 300.5(B) and NM cable is prohibited to be used in &quot;wet or damp locations&quot; by 334.12(B)(4)

300.5(B) Wet Locations. The interior of enclosures or raceways installed underground shall be considered to be
a wet location. Insulated conductors and cables installed in these
enclosures or raceways in underground installations shall be listed
for use in wet locations and shall comply with 310.10(C). Any
connections or splices in an underground installation shall be
approved for wet locations.
334.12(B) Types NM and NMS. Types NM and NMS cables shall not be used under
the following conditions or in the following locations:
(4) In wet or damp locations


3rd Problem: A run of 120' may have too much of a voltage drop for 8 awg conductors feeding 50 amps
Depending on the length of the branch circuits fed by the subpanel, even if the 8 awg conductor is rated for the highest allowable ampacity like THWN-2, the voltage drop may exceed 5% at the end of one or more circuits over a 120' length. This problem is not strictly a code violation to my knowledge since it is not a safety hazard (it comes from NEC guidance in the form of an informational note in 210.19(A) quoted below), but rather can adversely impact the performance of tools/appliances fed by the circuits with such a large voltage drop. For this reason, I would strongly recommend upsizing the conductors to at least 6 awg THWN with a 50A breaker or upgrade to 8 awg THWN and downsize to a 40A breaker if 40A suits your needs for this subpanel.

210.19 Conductors — Minimum Ampacity and Size.
(A) Branch Circuits Not More Than 600 Volts.
Informational Note No. 4: Conductors for branch circuits as defined in Article 100, sized to prevent a voltage drop
exceeding 3 percent at the farthest outlet of power, heating, and
lighting loads, or combinations of such loads, and where the maximum
total voltage drop on both feeders and branch circuits to the farthest
outlet does not exceed 5 percent, provide reasonable efficiency of
operation. See Informational Note No. 2 of 215.2(A)(4) for voltage
drop on feeder conductors.


TL;DR
There are several code violations and code is primarily developed to ensure safety. You may also notice suboptimal performance from whatever devices are hooked up to the outlets fed by this subpanel if you draw more than 40 A from the subpanel. It also seems like this work was done by someone who is unfamiliar with code; I wouldn't be surprised if there are other less obvious code violations and safety hazards. I would recommend calling in an electrician to rework this subpanel installation to meet your needs for it according code. I would also recommend calling in someone to inspect the other visible portions of the wiring (junction boxes/outlets) after the subpanel.