4 ohm speaker

The "50watts" part is the one we notice first and everything else qualifies how that "50watts" was measured. Having enough power is what most people look for in an amp. However, other things come into play. If the you are going to run a load less than 4 ohms, then the current capability of the amp is definitely important and most specs do not give a current capability. A power rating into 2 ohms can help though. If the power doubles into 2 ohms then you know that the amp is built strongly enough that it can deliver enough current to drive a 2 ohm load. You may think that this is not important if you are not going to drive 2 ohm loads but it is important. Speakers (woofers, midranges, tweeters, etc) are not purely resistive. They have capacitive and inductive properties as well. Depending on the music and your setup, the impedance may dip well below 4 ohms for a nominally 4 ohm speaker.
Whether you amp can supply current fast enough to reproduce the music faithfully depends partially on the amp's slew rate (how fast its output can change), its damping factor (how easily it can control the speaker) and its current capability. For these reasons 2 ohm power is important even when driving 4 ohm speakers. Slew rates of 100V/microsec and damping factors above 100 (referenced with a 4 ohm load) are good but that information is usually not given out by the amp manufacturer. I hope it is clear now that the number of watts an amp can produce is only one factor in determining whether an amp is capable of the performance you desire.
On a final note on this part of the spec, most head units use IC (integrate circuits or chips) for the built-in amp's output stage. Those chips rarely can provide adequate current which is why even most novices know not drive subwoofers from a head unit. Real amps often have ICs in them as well but the output stages are almost always discrete, meaning they are built from transistors, resistors, capacitors and not integrated together inside tiny ICs. Advances in IC technology always making them better though.