Defining a Competitive Equilibrium for a two Period Model
A competitive equilibrium is specific to the economy you're modeling. Your economy has three markets:


Capital market for $K$
Labor market for $L$
Goods market for $Y$


Furthermore, there are several optimization problems / budget identities that have to hold.


The household has a consumption/savings and a leisure/labor decision
The firm has a labor and a capital choice
You need to ensure that the government's budget constraint holds.


In a competitive equilibrium, you need to clear all the markets, and any quantity that is chosen, needs to be done so optimally, given everything that's happening in your economy. Here, you have two periods which need to be in equilibrium at the same time. If I were grading your homework, I would expect you to write the above for minimum points. You should then show what the conditions/equations are for each of the bullet points.

Example: Multi-Period Non-Equilibrium

For example, the household will chose savings in period 1, given what he expects to do with those savings in period 2. If you'd expect him to do something stupid with the savings in period 2 (say, throw them away), then it would be optimal to save nothing at all. In this case, you would be at an equilibrium in period 1. However, your choices for period 2 would be non-optimal, hence we do not have a 2-period equilibrium.