I have a set of predictor variables and a target variable. I am really
confused with regards to what method to use for forecasting the target
variable.
Start with the same things as you would started with analyzing this data as usual: look at the plots, summary statistics, clean the data if there are any errors, analyze the missing data -- if needed make decisions on what to do with missings (e.g. use single or multiple imputation). Think about your problem: What is your data? What do you want to know? Does your data enable you to answer the question that you are asking? If not, maybe you can rephrase your question to answerable one? Is there any pattern in your data that makes forecast possible (if it is purely random than your options are limited; search for "forecastability"). Consider if your data is sufficient for forecasting (e.g. if you want to make a prediction about next five years than you should have at least data on the previous five years, but in most cases much more than that). If you are modeling time-series than you have to thing about nature of the series: is there any seasonality (e.g. increases in summer and drops in winter)? Are there any things that happen with some regularity that influence your data? Is your data autocorrelated? Finally, do you have any a priori knowledge about your data (e.g. if you want to predict human height it simply cannot be lower than zero)? Take all those cases into consideration. You can find a friendly popular introduction to thinking about forecasting in Nate Silvers (2012) book The Signal and the Noise. See also The Black Swan by Taleb (2007) for critique and examples of forecasts going wrong.
Now, after spending some time with looking and thinking about your data you have to choose appropriate method or model for it. If it is time-series data than consider one of the multiple methods for modeling and forecasting time-series (e.g. exponential smoothing, ARIMA). You can include time component in regression or generalized linear model and in some cases this is preferable method. Sometimes you need non-linear models, machine learning methods or other. If you want to include out-of-data information in your model you may need a Bayesian model. You may be also interested in conducting simulation and then base your judgment based on possible scenarios that emerged from simulation. There is too many possible choices to summarize them in a single answer, so if you are not familiar with those methods than start with some statistics handbook, check also handbooks on time-series (e.g. Chatfield, 2003) and forecasting (e.g. Hyndman and Athanasopoulos, 2013). Notice also that sometimes simpler methods perform better than the complicated ones.
If you made your forecast, then you have to asses its performance. For this you can use bootstrap, cross-validation, hold out sample (sample that was not used during model training phase and is used only for testing your model), you can learn your model on first $N-k$ observations and try to predict results for the following $k$ cases. Remember that in most cases perfect forecast is not possible, you are looking for the best one you can get from this data and with the tools that you have. Remember also that it is often the case that if you take few forecasts made using different methods and take weighted average of them, then the averaged forecast ofter outperforms individual forecasts.
For example, my data set has monthly customer profit (which is my
target variable) and a set of predictor variables (balances of
different accounts) for one year for each customer.
It is hard to comment on this one, because it really depends on what is your data and what you want to forecast, but reviewing the literature should help you to get some insight about methods that fit your problem (check the links I provided and the books I refer to for some introduction).
I need to predict the profit for the next 5 years. I am confused in
that I do not have the data (predictor variables) for the future.
Well... that is what forecasting is about. You build a model using the data that you have and then use this data to make educated guesses about the future. Illustrating it with simple regression model, imagine that you have model
$$ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i $$
you use some data for estimating this model what leads to obtaining $\hat \beta_0$ and $\hat \beta_1$ parameters, next you use those estimated parameters and external data $x^*$ to predict unknown $y^*$ by using the formula
$$ y_i^* = \hat\beta_0 + \hat\beta_1 x_i^* $$
In this thread you can find example for making such predictions using a Bayesian model in JAGS. This part is tricky because you have to consider if it is really the case that model estimated on the data you have is adequate for applying it to the future (e.g. you have data on growth of 5-year-olds and want to use it to predict growth of adults -- the model would be obviously incorrect because rapid growth of children stops at some point). Remember that your forecast would probably be wrong, provide a prediction interval so to asses possible variability of the future values rather than single point estimate. Finally, remember that all models are wrong and you are looking for a useful one.
Silver, N. (2012). The Signal and the Noise: Why So Many Predictions Fail-but Some Don't. Penguin Group.
Chatfield, C. (2003). The Analysis of Time Series: An Introduction. Chapman and Hall/CRC.
Hyndman, R.J. and Athanasopoulos, G. (2013). Forecasting: principles and practice. OTexts.
Taleb, N.N. (2007). The Black Swan: The Impact of the Highly Improbable. Random House.
