Waves are created when vibrstion energy moves from one place to another. It will move at a speed or velocity equal to the distance it travels divided by the time it takes to get there. Because simple waves have a pattern that easily described and measured, we can count how often the vibrations happen, or how much time elapses between the vibrations, or how far the wave travels in one vibration, or how big the vibrations is.
The frequency of a wave is usually expressed in cycles per second, or Hertz. The time it takes for one vibration, the seconds per cycle, is the inverse of the frequency and is called period. A wave moving at a frequency of 2 Hz has a period of 0.5 seconds. A wave that vibrates or cycles often has a "high" frequency; one that has a large period has a "low" frequency.
The distance the energy travels in one cycle is called the wavelength. This can be in any units of distance: km or m for "long" waves; cm, mm or µm for "short" waves. Because it is a measurement of a pattern, it can be measured from a point on any part of the wave to the same point on the next same part of the wave. It is easiest to measure between two crests (high points) or two troughs (low points).
Amplitude measures how big the wave is. Depending on the type of wave it is, this can be measured in many ways with many different units. A water wave's amplitude would be measured with height: a 2-foot wave vs a 20-foot wave. A sound wave's amplitude might be measured with air pressure: a puff vs a boom. A light wave's amplitude might be measured with brightness: like a candle or like the sun.
The speed of a wave can be taken by dividing the distance (wavelength) by the time (period), but often it is more convenient to multiply the distance by the frequency (the inverse of the period). This also shows the important relationship that for waves going the same speed, if the wavelength goes up, the frequency goes down, and vice-versa. This is why instuments that make low notes are longer than instruments that make high notes.

When a wave comes from a mechanical vibration of an object, that wave requires a medium to travel. A medium is material that "carries" the wave. Sound is a mechanical wave and needs to travel through matter, whether it is gas, liquid or solid. An electromagnetic wave travels through its own electric and magnetic fields and does not require matter in order to travel. All forms of light are electromagnetic waves and they travel from stars through virtually empty space.

When a wave travels out from a vibration that is perpendicular to it, that wave is transverse. Transverse waves have crests and troughs. Examples are water waves, surface earthquake waves, and "doing the wave" at a sporting event.

When a wave travels out from a vibration that is back and forth along the same line, that wave is longitudinal. Longitudinal waves have compressions and rarefactions and are also called compression waves. The best example is sound waves. Other pressure waves and deep earthquake waves are longitudinal waves, too.

When waves travel through a medium and reach a boundary with a different medium, a few things can happen. The wave energy might be absorbed into the medium and become heat. If there's an opening, a gap, or a corner in the new medium, then diffraction can bend the wave out of that gap or around that corner. The wave could bounce back off the boundary. How the wave hits the boundary can help predict how the wave reflects off it. The wave can continue into the new medium and if it changes speed, then it will refract into a different direction. How it hits and how it changes speed can help predict how the wave will bend.
When a wave reflects, the angle at which it hits the boundary is the same as the angle when it leaves the boundary. When a water wave hits the side of a pool or bathtub or sink, or when light hits a mirror or something shiny, we measure the angle of incidence between the incoming wave vector and a line normal (perpendicular) to the boundary. The angle of reflection, between the reflected wave vector and the same normal line, will be the same amount as the angle of incidence.

When a wave refracts, the wave vector bends, so the angle at which it travels in the first medium is different from the angle at which it travels in the second medium. If the wave slows down, it bends towards the line normal (perpendicular) to the boundary. If the wave speeds up, it bends away from that normal line. If the boundary is a curved surface, then the light will converge or diverge like a lens.

The speed of a wave can be calculated by multiplying frequency and wavelength, but it depends primarily on the medium through which it travels. If the medium is more elastic or "bouncy", then a mechanical wave can vibrate through the material better. For this reason, waves move faster through solids than through liquids and faster through liquids than than through gases. Solids are more elastic because they try to retain a solid shape. Liquids can slip and splash, so they are less elastic. Gas molecules are not very connected to each other, so they don't bounce well.
If the medium is less dense, then a mechanical wave has more room to vibrate through the medium. For this reason, waves generally move faster in warmer media than in colder media. This is why instruments have to be tuned before a concert, because temperature (and humidity a bit) affects the exact sound they produce.

When the source of a sound moves, its frequency seems to change depending on whether it moves closer towards you or moves farther away from you. This apparent change in frequency when you or the source of sound moves closer together or farther apart is called the Doppler effect. Of course, there's also the separate fact that sounds that are farther away are quieter than those near you whether they are moving or not. When a siren on an emergency vehicle travels towards you, it gets louder but also sounds higher in pitch. When it moves away from you, it gets quieter but also gets lower in pitch. The siren itself sends out the same wave the entire time, but how it reaches you changes how you hear it. The Doppler effect is the effect of the frequency or pitch of the sound changing.