A continuous time model treats time as a continuous quantity which may take any value: i.e. units of time can always be divided into smaller units. This is in contrast to discrete time models which treat time as changing in jumps.
Although continuous time model are more realistic with regards to time, they also usually treat prices as continuous as well, which they are usually not: most securities prices move in ticks.
Discrete time models tend to be mathematically complex: to most mortals even discrete time models are complex, but continuous time models are more so. Using discrete time models (directly, or, to derive usable equations such as Black-Scholes) means solving differential equations with stochastic (random) terms. The stochastic term is typically some form of Brownian motion type price process.
Many models do not yield a closed form solution (i.e. an equation) and must be solved using computational methods: approximations that rely on (thankfully computerised) calculations. These are similar to (if more sophisticated) by calculating the area under a function by counting squares on graph paper, rather than by integrating and getting an exact answer.