Measurement-based quantum computation represents a powerful and flexible framework for quantum information processing, based on the notion of entangled quantum states as computational resources. The most prominent application is the one-way quantum computer, with the cluster state as its universal resource. Here we demonstrate the principles of measurement-based quantum computation using deterministically generated cluster states, in a system of trapped calcium ions. First we implement a universal set of operations for quantum computing. Second we demonstrate a family of measurement-based quantum error correction codes and show their improved performance as the code length is increased. The methods presented can be directly scaled up to generate graph states of several tens of qubits.