a four-vector

A three-dimensional vector like velocity is more than three numbers — it is a geometric arrow whose components reshuffle in a definite way when you rotate your axes, so that its length is unchanged. A four-vector is the spacetime upgrade: a set of four numbers (one 'time' component, three 'space') that transform into each other under a Lorentz boost exactly the way the coordinates (ct, x, y, z) do, so that the four-vector has an invariant 'length'.

Formally, a contravariant four-vector A^mu = (A^0, A^1, A^2, A^3) is an object whose components in a boosted frame are A'^mu = Lambda^mu_nu A^nu, where Lambda is the same Lorentz transformation matrix that acts on coordinates. Its invariant magnitude is A^2 = eta_munu A^mu A^nu = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2 — a single number all inertial observers agree on, positive for time-like four-vectors, negative for space-like, zero for null. The inner product of two four-vectors, A·B = eta_munu A^mu B^nu, is likewise invariant.

Four-vectors are the grammar of covariant relativity: writing a law as an equality between four-vectors (or tensors) guarantees it holds in every inertial frame automatically — this is 'manifest Lorentz covariance'. You will meet the position x^mu = (ct, x, y, z), the four-velocity, four-momentum, four-current, and four-potential, all built to transform this way. Caveat: not every list of four numbers is a four-vector — energy alone, or the three-velocity glued to a 1, does not transform correctly; you must check the transformation law.