Do they all mean to receive or to obtain? What are the differences in the use of these words?
- bekommen: means to get something. this is the most generic term in the group since it has the least implications of how and what. It implies only that the process is finished as opposed to "empfangen". Can be used for people "ich.. " and entities "Die Firma... ".
- erhalten: means to receive something that you have expected, waited for, mostly for objects. It could be a letter and the term "halten" (="hold") indicates that it usually is something that is small enough to hold in your hands. Therefore it does not work for something continuous such as a disease, a phone connection or like a radio signal. There must be a point when you have what is being transferred. Works for people and entities.
- entgegennehmen: means to take something that someone hands over to you in a more official manner, like for a registered letter. It implies that the object becomes your possession. The term "nehmen" (="take") indicates that you participate in the act of receiving something. Like erhalten, it does not work for something continuous. Works only for people.
- empfangen: means to receive something or someone that comes to you. It does not indicate how you got it (in your hands or not) or if you initiated the transfer, so it is used for emails, and can be used for something continuous like a Radio signal. The term "fangen" (="catch") indicates an action of placing/enabling yourself in order to receive it, like tuning a radio or going to the door for your guests. It does not work for something you passively receive like a disease or a snowball in your face.
- besorgen: means to go somewhere and get something, like shopping or if you went to get it from a friend. You initiate the transfer. It indicates that you fill a need and you can do that for someone else: "Ich habe meiner Mutter einen neuen Ofen besorgt".
- kriegen: means a passive receiving of something. It can indicate that you had little or no control in getting it, used for an injury for example. It could be taken as almost 100% equivalent to "bekommen" in a colloquial form.
Please note that bekommen and kriegen can both change the meaning with the appropriate preposition: Herausbekommen, hinkriegen, rauskriegen.
Another thing to consider is that a lot of idioms where a physical action is described have a combined meaning that is in the end not a physical action. Therefore, "Wie kriegen wir die Kuh vom Eis?" means literally "how to we get (=receive) the cow off the ice", as in physically get it off from somewhere, but figuratively means how to solve a problem.