One-dimensional nearest-neighbour random walks
A random walk is a random process consisting of a sequence of discrete steps of fixed length. In the case of one-dimensional nearest-neighbour random walks, the reachable states are integer (or natural) numbers and at each step, the process jumps to the nearest-neighbour to the right with probability p or to the nearest-neighbour to the left with probability 1 - p.
A random walk with state space equal to Z (integer numbers) can be transient or null recurrent depending on p. If p=1/2 then it is null recurrent and in other cases it is transient. Look at the book by S. Ross (Example 4.3d).
A random walk with reflectant barrier at 0 (i.e., state space equal to non negative integers) can be transient, null recurrent or positive recurrent. Look here (local copy).