"And if my subjects (servants) ask you about me, indeed I am close . . " (2:186)

The Month of Dua' (Supplication)

"Indeed, Dua' is worship itself" as stated in one Hadith (Bukhari and Ahmad). Thus a Muslim does Dua' / invocation and supplication, all the time, but usually more so in Ramadan, as Allah followed the verses of Ramadan with the one about Dua'. 
The Quran also made another powerful reference to Dua' in Surah Ghafir (40:60) "Your Lord has proclaimed, “Call upon Me, I will respond to you. Surely those who are too proud to worship Me will enter Hell, fully humbled.” thus, indicating that Dua' is worship.
Dua' can be of 2 types, asking Allah for specific needs, or just using supplication as a way to get closer to Allah and to humble the self. Ramadan is the best for both.
There are so many books and booklets on the subject, and well-known Dua's from the Sunnah of the prophet (PPBUH).