I thought both tremolo and vibrato are the two main ways of melody inflections on instruments. Yet I noticed singers only talk about vibrato with their voices and not tremolos, why is this?
As @JSheldon says, vibrato is varying pitch, tremolo is varying loudness.
Why singers vary pitch more agilely than loudness is an accident of human musculature -- songbirds are quite different, for instance. The muscles that adjust the tension of the vocal cords (to adjust pitch) are tiny, and can change tension rapidly. The diaphragm and other apparatus that pushes air past the vocal cords (changing air flow is the main cause of changing loudness) is larger, heavier, slower. You can fake a tremolo by tapping quickly on your chest like a gorilla (or on a three-year-old's, to amuse them).
A secondary cause of changing loudness, for a professionally trained singer, is shaping the mouth to align its resonances more or less closely to the pitch being sung (tuning a filter, in engineer-speak). Those muscles are again small and agile. If you inspect a spectrogram of an opera singer holding a long note, you'll see fast variation in both pitch and loudness.
In terms of the physics of sound, vibrato is usually defined as a variation of pitch, while tremolo is a variation of volume.
Basically, a singer's vibrato happens naturally when they are singing healthily with good technique. A well-trained/well-skilled singer can modify the pitch and speed of their vibrato, or sing with straight tone (without vibrato) when needed. The vocal world doesn't usually use the term tremolo because we don't vary the volume when natural vibrato occurs, just the pitch.
Vibrato results from a slight modulation back and forth in pitch, which is caused, in the context of the voice or a wind instrument, by modulating the volume and over/underblowing, per se, the note. In strings it's different, of course, but mostly what it boils down to is a. Vibrato and tremolo are tightly connected in singing and wind contexts, so doing one without the other is very difficult; b. Vibrato covers up bad pitch, which is convenient; and c. Vibrato was just what was used in the baroque/classical period and it stuck. Regardless of the most dominant factor, vibrato and tremolo go hand in hand in any musical application that involves the lungs, and have therefore not tended in the past to be considered especially different in such fields. I would be surprised, however, if no contemporary piece has been written specifying a vocalist to use tremolo sans vibrato. Sounds like something Roomful of Teeth would do...
Tremolo is playing a note repeatedly very quickly. Violinists (and fiddle players) do this by moving the bow back and forth very quickly (think Charlie Daniels' playing long sustained sliding tremolos in "The Devil Went Down to Georgia"). Mandolinists and guitarists do this by rapidly picking a note with a plectrum or fingers. A pianist commonly does a tremolo on octaves by repeatedly striking the keys.
A vocalist might do this by using consonants to start a note over and over, like "ta ta ta ta ta ta ta" or "da da da da da da" or by using a fluttering or rolling effect with the tongue. A vocalist could also create a tremolo by opening and closing the glottis, or by using the diaphragm to vibrate the vocal chords with puffs of air, like "ha ha ha ha ha".
In short, tremolo for vocalists is a bit of a "weird" technique, and likely is only called for in avant garde classical music.
Vibrato is a slight, smooth wavering of the pitch — and a more common vocal technique for dominant western styles of jazz, classical, rock and country. 1970s Country & Western singer Tanya Tucker had a very fast vibrato, which almost simulates a tremolo sound. Some people referred to her sound as "nanny goat" vibrato. But far more common is a smoother style of vibrato found in a singer like Andrea Bocelli performing Con te partirò.