Short answer
Dua means calling on Allah: asking Him for what you need, in any language, at any time, for this life and the next. The Prophet ﷺ said dua is worship itself. Allah promises in the Quran to answer whoever calls on Him, and dua is made to Allah alone.
Dua is mustahabb at all times and sunnah at its set times, such as after salah. Directing dua to anyone other than Allah, asking them for what only Allah can give, is shirk.
The four schools agree on this.
Explanation
The Arabic du'a comes from a root meaning to call or to invite. In Islam it means turning to Allah and asking Him. It is different from salah, the five set prayers with fixed words and movements. You can make dua in salah, after it, or anywhere else, in Arabic or your own language: see dua in English.
The Prophet ﷺ said, "Dua is worship," and then recited, "Call upon Me; I will respond to you" (40:60). Asking Allah shows that you need Him and that He alone can give, which is the heart of worship. That is why dua is only made to Allah. Asking a living person to pray for you is fine; see tawassul.
Two kinds of dua: asking (du'a al-mas'ala), such as asking for health or forgiveness, and praise (du'a al-ibada), such as glorifying Allah, which is also a form of calling on Him. Many of the best duas are in the Quran itself, such as Rabbana atina.
Allah says He is near and answers the caller (2:186). For the manners of dua see how to make dua properly; for the best times see when duas are accepted; and if you feel your duas go unanswered, see why is my dua not answered. The four schools agree on all of this. You can find duas for daily life in the dua collection.
The evidence
Quran, Surah Ghafir 40:60
وَقَالَ رَبُّكُمُ ٱدْعُونِىٓ أَسْتَجِبْ لَكُمْ ۚ إِنَّ ٱلَّذِينَ يَسْتَكْبِرُونَ عَنْ عِبَادَتِى سَيَدْخُلُونَ جَهَنَّمَ دَاخِرِينَ ﴿٦٠﴾
And your Lord says, "Call upon Me; I will respond to you." Indeed, those who disdain My worship will enter Hell [rendered] contemptible.
Allah commands dua and promises to answer it.
Quran, Surah Al-Baqara 2:186
وَإِذَا سَأَلَكَ عِبَادِى عَنِّى فَإِنِّى قَرِيبٌ ۖ أُجِيبُ دَعْوَةَ ٱلدَّاعِ إِذَا دَعَانِ ۖ فَلْيَسْتَجِيبُوا۟ لِى وَلْيُؤْمِنُوا۟ بِى لَعَلَّهُمْ يَرْشُدُونَ ﴿١٨٦﴾
And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided.
Allah is near to the one who calls on Him.
Jami' at-Tirmidhi 2969
Grading to be checked... الدعاء هو العبادة ...
"Dua is worship."
Dua is itself an act of worship, owed to Allah alone.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.