akzeptieren usually feels a little more distant than annehmen. "Er akzeptiert mich" can be used even if he only begrudgingly accepts you, while "er nimmt mich an" connotes that he takes you in his heart. "Eine Entschuldigung annehmen" feels more like reconciliation than "eine Entschuldigung akzeptieren". So in general, akzeptieren is more formal, neutral, reserved, while annehmen connotes that you do it gladly or at least don't have stomach cramps about it.
If you start a new job, you would say "Ich nehme eine (neue) Arbeit an". It's not the affectionate kind of "annehmen" I described above, though, it just means you take the job.
"Ich akzeptiere den Job" means someone offers you to do a job/ task, and you accept the conditions.
The suffix ‘-ieren’ comes from a sort of misunderstanding of the stem of certain Latinate loanwords as including the infinitive endings (‘-are/-ere/-ire’), a process known as ‘rebracketing’. It was subsequently applied to many verbs borrowed from Latin (and Romance languages, e.g. ‘telefonieren’, ‘spazieren’, ‘sortieren’, ‘kapieren’) and eventually also used as a sort of causative suffix for native German words too (similar to English ‘-ise/-ize’, ‘-(i)fy’, or ‘-en’, as in ‘lessen’ or ‘soften’), like in ‘möblieren’ (‘to furnish’, from ‘Möbeln’, ‘furniture’) or ‘halbieren’ (‘to halve’, from ‘halb’, ‘half’).
It has to do with one of the 4 German cases. In this example, "Ich" is in the accusative case. This case, as my textbooks tell me, says that the direct object of the sentence is always put in this case. The direct object being the person or thing directly affected by the action of the verb.
In addition, the word "mir" is in the dative case. This case states that, the indirect object of the sentence is always in the dative and that the indirect object is the person or thing that receives the direct object.
For example, in "Der Junge gibt dem Hund das Essen" the dog is the indirect object receiving the food from the boy and the food is the direct object because it is the thing that's being given or affected by the verb.
The other two cases are the Nominative and Genitive.