Takeem (also spelled taqasim, taqsim, takseem, taxim, etc, etc) is a solo melodic improvisation. There is a structure to it that explores the maqam the taxeem is in and returns to the home note established at the start. There is more to it that gets pretty technical, for now you should just be aware that it is the musician following their feeling and displaying artistry. One interesting thing, it means "division" in Arabic, and something along the lines of "bridge/connection" in Turkish. It is often used to create a smooth transition from one song to another, especially if there is a maqam (mode/scale/key) change from one song to the next.
It can be performed entierly solo,
over a drone note,
or over a simple rhythm with few heavy doums, like wahida, chiftitelli, or masmoudi kabir.
An accordion or, less frequently, a violin taxeem often preceeds a beledi progression played for women.
The mawaal is the vocal version of a takseem. It can mean doing vocal runs around lyrics, or just improvising singing on an abstract phrase like "ya leyl, ya ainy". It can be found in formal-artistic music, pop music, or folk music. This phrase mean "ph night, oh my eye" and is a bit like improvising a tune with the sound "lalala" in English.
This clip is Om Kalthoom, a national treasure of Egypt and widely regarded as one of the best singers in the world, performing a mawaal.
The clip of Adeweya his mawaal starts at about 2 minutes in. He is credited with starting the shaabi boom of the cassette era, which of course paved the way for the modern festival music that is now so popular.
Tarab is the goal of many types of music, especially art music. It loosely translates to "the ecstasy of listening" and is a transcendent state achieved by being in tune with the music and the moment. I strongly encourage you to read this short blog entry from a language learning site to learn a bit more. It is similar to duende in flamenco, or the transcendent state sought in certain jazz stylings.