Categories of Actions in Islam
Question

Fard (obligatory), Mustahabb (encouraged, not obligatory), Mubah (permissible, optional), Makruh (not encouraged, disliked), Haram (prohibited) – I hope that you can give me an example of each of these categories.
Praise be to Allah.There are 5 categories of actions in Islam and they are as follows:

Obligatory (Wajib, Fard ), is that which the Lawgiver (Allah) has enjoined by way of it being compulsory. 

Examples of obligatory actions include the five daily prayers, fasting Ramadan, Zakah for those who are obliged to give it, and Hajj to the Sacred House for those who have the means of doing so. 
The one who does it will be rewarded for obeying the command, and the one who does not do it deserves to be punished. 

Encouraged or recommended (Mustahab, Sunnah, Masnun or Nafl ), is that which is prescribed by the Lawgiver, but not by way of it being obligatory or compulsory. 

Examples of recommended actions include Qiyam Al-layl, the regular Sunnah prayers that are additional to the five obligatory prayers, fasting three days of every month, fasting six days of Shawwal, giving charity to the poor, and regularly reciting Adhkar and Quran. 
The one who does it will be rewarded for complying, but the one who does not do it will not be punished. 

Prohibited or forbidden (Haram), is that which the Lawgiver prohibits in the sense that it is obligatory to refrain from doing it. 

Examples of forbidden actions include fornication or adultery, Riba (usury), drinking alcohol, disobedience to parents, shaving off the beard, and women displaying their adornment in front of men (who are not allowed to see her without Hijab). 
The one who refrains from doing that which is prohibited will be rewarded for complying, and the one who does it deserves to be punished. 

Disliked (Makruh), is that which the Lawgiver disallowed, but not in the sense of it being obligatory to refrain from it. 

Examples of disliked actions include giving and taking with the left hand; women following funeral processions; socialising after the `Isha’ prayer; praying in a garment of which no part is covers the shoulder; offering supererogatory prayers after Fajr until the sun is fully risen, or after `Asr until the sun has set. 
The person who refrains from a disliked action will be rewarded for complying, but the one who does it will not be punished. 

Permissible (Mubah, Halal, Ja’iz), is that to which no command or prohibition is connected to the deed itself. 

Examples of permissible actions include eating and drinking; buying and selling; travelling for the purpose of tourism or seeking provision; engaging in intimacy with spouses during the night in Ramadan. 
The restriction on the definition of what is permissible is indicated by the phrase “To the deed itself”, because there may be an instruction that is connected to it, which makes it enjoined or disallowed. 
In principle, buying water is permissible, but if doing Wudu for an obligatory prayer depends on that, then it becomes obligatory to buy it, because anything without which an obligatory duty cannot be completed becomes obligatory. 
In principle, travelling for the purpose of tourism or leisure is permissible, but if that travel is to a land of the disbelievers in which there is a great deal of temptation and evildoing, and immorality is widespread, then that travelling becomes prohibited, because it is a means that leads one to falling into that which is unlawful. 
For more details, please see the following answers: 174947 , 26242 , 14258 , and 36546
For more information, please see the following books:

Rawdat An-Nazir wa Jannat Al-Manazir by Ibn Qudamah (1/150-210)
Al-Bahr Al-Muhit by Az-Zarkashi (1/140-240)
Sharh Al-Usul min ‘Ilm Al-Usool by Ibn ‘Uthaymin (p. 46-68) 

And Allah knows best.