From BiblePortal Wikipedia

Vine's Expository Dictionary of NT Words [1]

1: Βασίλειος (Strong'S #934 — Adjective — basileios — bas-il'-i-os )

from basileus, "a king," is used in  1—Peter 2:9 of the priesthood consisting of all believers. Cp.   Luke 7:25 , for which see Court , No. 3. In the Sept.,  Exodus 19:6;  23:22;  Deuteronomy 3:10 .

2: Βασιλικός (Strong'S #937 — Adjective — basilikos — bas-il-ee-kos' )

"belonging to a king," is translated "royal" in  Acts 12:21;  James 2:8 . See King B No. 2 Nobleman

Webster's Dictionary [2]

(1): ( a.) Kingly; pertaining to the crown or the sovereign; suitable for a king or queen; regal; as, royal power or prerogative; royal domains; the royal family; royal state.

(2): ( n.) An old English coin. See Rial.

(3): ( a.) Under the patronage of royality; holding a charter granted by the sovereign; as, the Royal Academy of Arts; the Royal Society.

(4): ( n.) Printing and writing papers of particular sizes. See under paper, n.

(5): ( n.) A small sail immediately above the topgallant sail.

(6): ( n.) One of the upper or distal branches of an antler, as the third and fourth tynes of the antlers of a stag.

(7): ( n.) A small mortar.

(8): ( n.) One of the soldiers of the first regiment of foot of the British army, formerly called the Royals, and supposed to be the oldest regular corps in Europe; - now called the Royal Scots.

(9): ( a.) Noble; generous; magnificent; princely.

(10): ( n.) A royal spade.

King James Dictionary [3]

ROY'AL, a. L. regalis, from rex, king. See Reck and Right.

1. Kingly pertaining to a king regal as royal power or prerogative a royal garden royal domains the royal family. 2. Becoming a king magnificent as royal state. 3. Noble illustrious.

How doth that royal merchant, good Antonio?

ROY'AL, n.

1. A large kind of paper. It is used as a noun or an adjective. 2. Among seamen, a small sail spread immediately above the top-gallant-sail sometimes termed the top-gallant-royal. 3. One of the shoots of a stag's head. 4. In artillery, a small mortar. 5. In England, one of the soldiers of the first regiment of foot, called the royals, and supposed to be the oldest regular corps in Europe.

International Standard Bible Encyclopedia [4]

roi´al  : Either belonging to a king (kingdom) or having kingly power, dignity, authority, etc. In Hebrew, the word is expressed by using different nouns in the gen. case (the "construct state"). They are: (1) melekh , "king": "Asher ... shall yield royal dainties," literally, choice morsels of the king, meaning fit for a king (  Genesis 49:20 ); "besides that which Solomon gave her of his royal bounty," literally, which he gave her according to the hand (the wealth) of King Solomon ( 1 Kings 10:13; compare the Revised Version margin); "a royal statute," literally, statute of a malkā' , which is the emphatic Aramaic term for melekh , "king" ( Daniel 6:7 ); (2) mamlākhāh , "the power and dignity of a king," "Gibeon ... one of the royal cities," i.e. a capital city with a king of her own ( Joshua 10:2; compare  1 Samuel 27:5 ); "all the seed royal," literally, the seed of the kingdom ( 2 Kings 11:1; compare  2 Chronicles 22:10 ); (3) malkhūth , "kinghood," "kingdom": "royal majesty," literally, majesty of kinghood ( 1 Chronicles 29:25 ); quite frequently in the Book of Esther; royal wine ( Esther 1:7 ); crown ( Esther 1:11; compare  Esther 2:17;  Esther 6:8 ); commandment ( Esther 1:19 ); "her royal estate," literally, her kinghood ( Esther 1:19 ); house royal ( Esther 2:16; compare  Esther 5:1 ); royal apparel ( Esther 5:1; compare  Esther 6:8 , 15); throne ( Esther 5:1 ); (4) melūkhāh , "kingdom," "kingly power and dignity": "royal city," literally, the city of the kingdom, meaning here that part of the city (Rabbah) in which the royal palace was situated ( 2 Samuel 12:26 ); "royal diadem," literally, turban of kinghood ( Isaiah 62:3 ); (5) in  Jeremiah 43:10 we find the word shaphrı̄r  ; its meaning is uncertain: "royal pavilion" (the Revised Version (British and American) and the King James Version), "glittering" (Revised Version, margin), "scepter," "a carpet covering a throne."

The New Testament uses the word for basilikós , "belonging to king": "royal apparel" (  Acts 12:21 ); "the royal law," something like "the golden rule," being foremost because including all others ( James 2:8 ), and for bası́leios (being vested with kingly power and honor), "royal priesthood," the Hebrew rendering would be mamlekheth kōhănı̄m , "a kingdom of priests," i.e. a kingdom whose citizens are priests, emphasizing the two facts that the true Christians have free access to the grace of God and that they enjoy the liberties and privileges of His kingdom ( 1 Peter 2:9 ).