Makharij (مخارج) comes from the word kha(خ)-ra(ر)-ja((ج) which means to exit. So its how breath leaves your lungs to make the letter sounds. We have vowels or Horoof Madiyah which are made with a completely open track and these are Alif (ا), Wow (و) and Ya (ي) and all other letters are made by focusing the sound in certian places and restricting airflow. The bulk of the letters are labiodental meaning they use the lips and the teeth. The actual motion to produce the sound is called the Sifaat, e.g. some have Ghunnah/Nasalation like the Meem (م) and Noon (ن) or Aspirations like Ha(ه) or Ta(ت) or Shaa(ش) where a small breath leaves the mouth after the pronounciation of the letter. Getting Makharij right is the first step to tajweed; you can learn this on your own through some great youtube videos or schedule a class with a Qari using the form Below.
The Word Haraakat (حركات) technically means movements, but in terms of language it modifies the letter with one of the Madiyah/Vowel sounds mentioned above. Each letter can be adorned with each vowel making the total combination of sounds to be 29 x 3 = 87. In addition the letter could have a sakoon (little 'o' on top of the letter to denote silent), shaddah (little 'w' on top and means pronouncing the sound once as silent and again with its vowel sound) or tanween (doubling up on the vowel sound and sounds like putting a silent noon(ن). There are a lot more adorners, some specific to the script you're reading. Knowing all the Adornments is the second step to proper recitation of the Quran.
This is the hardest part of Tajweed, there are 28 rules in total to proper pronounciation but in theory it is 6 major ones with a bunch of exceptions for certian letters and circumstantial changes. The 6 major ones are Idhar, Idgham, Aqlab, Ikhfa, Qalqalah and Madd wal Woquf.
Idhar: means to show, states that if Noon Sakin or Tanween is followed by any of the 6 letters (ء ه ع ح غ خ) then there will be an no resting, which means that the sound of noon will not be hidden. In other words instead of resting on the noon a person will read it normally.
Idgham: means to merge, states if Noon Sakin or Tanween is followed by any of the 4 letters (ي م و ن) from Yermaloon group with shaddah over it then the sound of Noon Sakin or Tanween should be merged with the letter while resting on it and without resting with 2 letters (ر ل) of Yermaloon group.
Aqlab: The rule states that if noon sakin or tanween is followed by the letter ‘ب’ then the sound of Noon sakin will be substituted with the sound of Meem sakin and nasal sound. Note this rule applies within a single word or two words side by side.
Ikhfa: which means to hide, is a very common rule. The rule states that if anywhere Noon Sakin or Tanween is followed by any of the 15 letters ( ت ث ج د ذ ز س ش ص ض ط ظ ف ق ك ) then the sound of Noon Sakin or the sound of Tanween will be hidden while resting on it. The resting should be long enough so that it is clear from the recitation that a person is resting on the ن.
Qalqalah: which means to echo, is applied whenever the following five letters (ب ج د ط ق) are silent (having sakoon on top of it). The rule states the letter is supposed to be bounced or echoed. The echo will be hard, therefore said with more force, when there is a shaddah on the letter, little lighter when an ayah (verse) ends with the letter, and finally more lighter when it is within the middle of an ayah.
Madd wal Woquf: means elongation and stopping. This works in conjunction with specific adornments to tell you how long certian letters and sounds should be stretched and when you can or can't take stops in recitation.
We actually don't recommend learning the rules as grammar through a class but through practice. Start listening to and reciting with known Qaris to build up the subconcious adherence to the rules, otherwise it may be overwhelming. Once you master Tajweed, you're 1/3 the way to being a Qari.