With a properly designed VoIP network, the audio quality will be equal to that of standard telephony using the PSTN. In fact, the voice quality in a VoIP network has the potential to be much better that that of standard telephony using the PSTN. For traditional telephony, that voice is band limited between 300Hz and 3400Hz. With VoIP, wideband codecs can be used which provide superior voice quality. For some examples you can access: http://www.globalipsound.com/demo/tutorial.php.
Regarding quality of service (QoS), many companies that have a single site that deploy a VoIP system do not bother to implement it and the voice quality is excellent. Note that a single VoIP call using the G.711 codec only requires about 87Kbps upstream and downstream (including the overhead of Ethernet and IP headers) on a 100Mbps link.
When routing calls over a WAN between sites, QoS should be used because the bandwidth between sites is usually quite limited. QoS should be assured at both Layer 2 (e.g, Ethernet, ATM, Frame Relay) and at Layer 3 (IP). For Ethernet, the standard is IEEE 802.1p. This standard allows for up to 8 different priority levels to be configured. For IP, Differentiated Services (Diffserv) is typically used. Diffserv is defined in IETF RFCs 2474 and 2475. When switches and routers that support QoS receive such marked packets, they are placed in priority queues.
Now, QoS is not idiot proof. For example if you have enough bandwidth between two sites to support X calls and you try to route the voice traffic of X+1 calls along that path, the voice quality will become horrible for everyone. For this reason, IP PBX manufacturers allow for the setting of bandwidth limitations when calls are routed specific locations.
When routing IP calls over the Internet, it is much harder to assure the quality of service. Even though you mark packets with a certain priority marking, that marking must be honored at every hop from source to destination otherwise there is the potential for the quality to degrade. This is the case where people can have the quality of a VoIP call drop below what it is on the PSTN. This does not happen all of the time, but rather when the network is congested.
There are multiple standards for measuring speech quality on VoIP connections. The latest standard is called PESQ and it is published in ITU standard P.862. You can get more information from http://www.pesq.org/. There are many test equipment vendors that provide solutions using this algorithm.