Time Series analysis is the study of sequences of random variables that are indexed by time. Random variables that are close together in time tend to be more closely related than those that are far apart in time, and thus they cannot be treated as independent as is done for many statistical techniques. With minor modifications, the methods of time series analysis can also be applied to spatially related random variables. Examples of real data that can be modeled as time series are stock prices, weather, and sales data. It's easy to think of more examples, which is why time series methods are so important.
Recommended Books

Time Series Analysis and Its Applications: With R Examples
Robert H. Shumway and David S. Stoffer
Key Features
 Intext exercises
 Errata
Key Topics
 ARIMA Models
 Autocorrelation
 Bayesian State Space Models
 Estimation
 Exploratory Data Analysis
 Filtering
 Forecasting
 GARCH Models
 Hidden Markov Models
 Large Sample Theory
 Linear Filters
 Nonparametric Spectral Estimation
 Smoothing
 Spectral Analysis
 Spectral Domain Theory
 Spectral Estimation
 State Space Models
 Time Domain Theory
 Unit Root Testing
Description
Although there are more comprehensive time series books out there (such as our other recommendation), this is our favorite introductory time series book. There are many code examples in R with a relatively small number of dependencies, and the exposition is very clear. Unless your interests lie strongly along finance lines, we recommend using this text.

Analysis of Financial Time Series
Ruey S. Tsay
Key Features
 Intext exercises
 Errata
Key Topics
 AR Models
 ARCH Models
 ARMA Models
 BlackScholes Pricing Formulas
 Conditional Heteroskedastic Models
 Continuous Time Models
 Duration Models
 Estimation
 Extreme Value Theory
 Forecasting
 GARCH Models
 HighFrequency Data Analysis
 Ito Process
 Jump Diffusion Models
 Kalman Filter
 MA Models
 Markov Chain Monte Carlo
 Models for Price Changes
 Multivariate Time Series Analysis
 Multivariate Volatility Models
 Nonlinear Models
 Principal Component Analysis and Factor Models
 RiskMetrics
 Seasonal Models
 State Space Models
 Stationarity
 Value at Risk
 Vector Autoregressive Models
 Wiener Process
Description
The focus of this book is financial time series, but the methods it contains are mostly generic time series methods. However, most of the exercise data relates to finance, and some of the chapters are specific to finance. It is fairly comprehensive as introductory time series books go. The major downside of this book is that a lot of the exercises involve R packages which obscure what is actually being calculated. You can dive into the source code of course, but it would be nice if the authors had exercises that were doable
from scratch.
Of course, this downside could be considered an upside in that it familiarizes you with R's time series package ecosystem.