For this causative situation, the two examples you gave can be referred to as 'receiver only' and 'receiver + agent' respectively, where the receiver is a person or thing being acted upon and the agent is a person or thing being made to act.
- subject + faire + infinitive + receiver
In this situation, the subject only has something happen to the receiver.
- "Je fais laver la voiture" - "I am having the car washed."
- "Je fais faire du café" - "I am having some coffee made"
Receiver + Agent:
- subject + faire + infinitive + receiver + par/à + agent
In this situation, the subject has the agent do something to receiver.
- "Je fais laver la voiture à/par Michel" - "I am having the car washed by Michel" or "I am having Michel wash the car".
- "Je fais faire du café à/par Michel" - "I am having some coffee made by Michel" or "I am having Michel make some coffee".
There is a third situation that I would like to bring to your attention though:
- subject + faire + infinitive + agent
In this situation the subject makes the agent do something, but not to anything else, as is there is no receiver.
- "Je fais manger Michel" - "I am making Michel eat"
- "Il fait partir l'homme" - "He is making the man leave"
The first 'faire' is conjugated based on the subject and tense. You don't necessarily need a second 'faire' unless this is the verb you wish to use for the action being done. In my examples, I provided both situations.
Some examples relating to tense:
Passé composé: "J'ai fait laver la voiture" - "I had the car washed"
Futur proche: "Je vais faire laver la voiture" - "I am going to have the car washed"
A complete analysis of a 'receiver + agent' sentence:
- "Il fait laver la voiture à/par Luc" - "He is having the car washed by Luc" or "He is having Luc wash the car"
- Il - subject
- fait - conjugation of 'faire', 3rd person present.
- laver - infinitive
- la voiture - receiver
- à/par - prepositions
- Luc - Agent
More examples and contexts can be found at: http://french.about.com/od/grammar/a/causative.htm
EDIT: Important comment below on the nuances on which prepositions to use.
There's a nuance for à and par. À is rather colloquial as it often
conotes that the task is inflicted to someone. Par is the standard
preposition for an agent, the focus is then on the task being done not
on the agent performing the task. Je fais lecher mes bottes à Michel
is humiliating; je fais lecher mes bottes par Michel is rather stupid…
– Stéphane Gimenez