I always thought that God’s providence means that He will provide. Apparently I got that wrong, probably in part because the word provide is within the word providence.

In the book Trusting God by Jerry Bridges, he quotes J.I. Packer defining it as:

The unceasing activity of the Creator whereby, in overflowing bounty and goodwill, He upholds His creatures in ordered existence, guides and governs all events, circumstances, and free acts of angels and men, and directs everything to its appointed goal, for His own glory.

Jerry Bridges himself has developed a more succinct version:

God’s providence is His constant care for and His absolute rule over all his creation for His own glory and the good of His people.

For further reference and Scripture:

Easton’s 1897 Bible Dictionary
Providence
literally means foresight, but is generally used to denote God’s preserving and governing all things by means of second causes (Ps. 18:35; Ps. 63:8; Acts 17:28; Col. 1:17; Heb. 1:3). God’s providence extends to the natural world (Ps. 104:14; Ps. 135:5-7; Acts 14:17), the brute creation (Ps. 104:21-29; Matt. 6:26; Matt. 10:29), and the affairs of men (1 Chr. 16:31; Ps. 47:7; Prov. 21:1; Job 12:23; Dan. 2:21; Dan. 4:25), and of individuals (1 Sam. 2:6; Ps. 18:30; Luke 1:53; James 4:13-15). It extends also to the free actions of men (Ex. 12:36; 1 Sam. 24:9-15; Ps. 33:14-15; Prov. 16:1; Prov. 19:21; Prov. 20:24; Prov. 21:1), and things sinful (2 Sam. 16:10; 2 Sam. 24:1; Rom. 11:32; Acts 4:27, 28), as well as to their good actions (Phil. 2:13; Phil. 4:13; 2 Cor. 12:9-10; Eph. 2:10; Gal. 5:22-25). As regards sinful actions of men, they are represented as occurring by God’s permission (Gen. 45:5; Gen. 50:20. Comp. 1 Sam. 6:6; Ex. 7:13; Ex. 14:17; Acts 2:3; Acts 3:18; Acts 4:27-28), and as controlled (Ps. 76:10) and overruled for good (Gen. 50:20; Acts 3:13). God does not cause or approve of sin, but only limits, restrains, overrules it for good. The mode of God’s providential government is altogether unexplained. We only know that it is a fact that God does govern all his creatures and all their actions; that this government is universal (Ps. 103:17-19), particular (Matt. 10:29-31), efficacious (Ps. 33:11; Job 23:13), embraces events apparently contingent (Prov. 16:9, 33; Prov. 19:21; Prov. 21:1), is consistent with his own perfection (2 Tim. 2:13), and to his own glory (Rom. 9:17; Rom. 11:36).

Webster
3. In theology, the care and superintendence which God exercises over his creatures. He that acknowledges a creation and denies a providence, involves himself in a palpable contradiction; for the same power which caused a thing to exist is necessary to continue its existence. Some persons admit a general providence, but deny a particular providence, not considering that a general providence consists of particulars. A belief in divine providence, is a source of great consolation to good men. By divine providence is often understood God himself.