Usually, vector and array are the same thing and are used interchangeably when programming. That is in most places, so you shouldn't worry much about it most of the time.
That said, language is imprecise and we sometimes have words that have different meanings in different contexts that end up meaning the same thing, or one word having two or more different meanings.
In C an array is a series of data elements of the same type that are stored in memory contiguously. Java inherited that meaning. It's not the same as a List or a Vector, as arrays are more basic than those.
A vector is a mathematical construct defined as an object that can be added to another object of the same type, or be multiplied by any real (or complex) number resulting in another vector of the same type. It also has a few other properties that are very useful.
A vector has dimension. It's the minimum number of different vectors that have to be combined to get every possible vector of that type. Velocity and acceleration are 3-dimensional vectors because space has 3 base directions and by adding multiples of those three base directions you can get any position in space. Position in a plane is a 2-dimensional vector, and individual numbers are 1-dimensional vectors.
One way to represent a vector of dimension n is by using an array of n elements each one representing how much of each base vector you have to add to get to your vector.
Since you can represent a vector using an arra of elements, with time, the two concepts were equated. So, in many places, they simply are the same thing and in some languages arrays are called vectors.
Another case where one word has two different meanings is, for instance, dimension. The wikipedia definition of a vector as a 1-dimensional array is an example. You are not talking of vector dimension here. You are talking about the computer representation of the data structure. A 3-Dimensional vector can be represented by a "1-dimensional" array of size 3. 3 numbers in line. A 3x3 matrix can be represented by a "2-dimensional" array, which is what programmers call an array of arrays. Yet a 3x3 matrix is also a mathematical vector of dimension 9 (since it has all the properties of a vector) and has 9 numbers. Confusing, huh?
Anyway, I think the answer is simply: don't worry. It all depends on context. The two words have different origins, but in the context of data structures, when they say vector they mean exactly array.