How can put multiple plots side-by-side in shiny r?

So it is a couple years later, and while the others answers - including mine - are still valid, it is not how I would recommend approaching it today. Today I would lay it out using the grid.arrange from the gridExtra package.

• It allows any number of plots, and can lay them out in a grid checkerboard-like. (I was erroneously under the impression splitLayout only worked with two).
• It has more customization possibilities (you can specify rows, columns, headers, footer, padding, etc.)
• It is ultimately easier to use, even for two plots, since laying out in the UI is finicky - it can be difficult to predict what Bootstrap will do with your elements when the screen size changes.
• Since this question gets a lot of traffic, I kind of think more alternative should be here.

The cowplot package is also worth looking into, it offers similar functionality, but I am not so familiar with it.

Here is a small shiny program demonstrating that:

library(shiny)
library(ggplot2)
library(gridExtra)

u <- shinyUI(fluidPage(
  titlePanel("title panel"),
  sidebarLayout(position = "left",
      sidebarPanel("sidebar panel",
           checkboxInput("donum1", "Make #1 plot", value = T),
           checkboxInput("donum2", "Make #2 plot", value = F),
           checkboxInput("donum3", "Make #3 plot", value = F),
           sliderInput("wt1","Weight 1",min=1,max=10,value=1),
           sliderInput("wt2","Weight 2",min=1,max=10,value=1),
           sliderInput("wt3","Weight 3",min=1,max=10,value=1)
      ),
      mainPanel("main panel",
          column(6,plotOutput(outputId="plotgraph", width="500px",height="400px"))
))))

s <- shinyServer(function(input, output) 
{
  set.seed(123)
  pt1 <- reactive({
    if (!input$donum1) return(NULL)
    qplot(rnorm(500),fill=I("red"),binwidth=0.2,main="plotgraph1")
    })
  pt2 <- reactive({
    if (!input$donum2) return(NULL)
    qplot(rnorm(500),fill=I("blue"),binwidth=0.2,main="plotgraph2")
  })
  pt3 <- reactive({
    if (!input$donum3) return(NULL)
    qplot(rnorm(500),fill=I("green"),binwidth=0.2,main="plotgraph3")
  })
  output$plotgraph = renderPlot({
    ptlist <- list(pt1(),pt2(),pt3())
    wtlist <- c(input$wt1,input$wt2,input$wt3)
    # remove the null plots from ptlist and wtlist
    to_delete <- !sapply(ptlist,is.null)
    ptlist <- ptlist[to_delete] 
    wtlist <- wtlist[to_delete]
    if (length(ptlist)==0) return(NULL)

    grid.arrange(grobs=ptlist,widths=wtlist,ncol=length(ptlist))
  })
})
shinyApp(u,s)

Yielding:

Using @Mike Wise example, you can also use splitLayout(cellWidths = c("50%", "50%")...to show two plots side by side.

ui..R

library(shiny)

shinyUI(fluidPage(
  titlePanel("title panel"),

  sidebarLayout(position = "left",
                sidebarPanel("sidebar panel",
                             checkboxInput("do2", "Make 2 plots", value = T)
                ),
                mainPanel("main panel",
                          fluidRow(
                            splitLayout(cellWidths = c("50%", "50%"), plotOutput("plotgraph1"), plotOutput("plotgraph2"))
                          )
                )
  )
)
)

server.R

shinyServer(function(input, output) 
{
  set.seed(1234)
  pt1 <- qplot(rnorm(500),fill=I("red"),binwidth=0.2,title="plotgraph1")
  pt2 <- reactive({
    input$do2
    if (input$do2){
      return(qplot(rnorm(500),fill=I("blue"),binwidth=0.2,title="plotgraph2"))
    } else {
      return(NULL)
    }
  })
  output$plotgraph1 = renderPlot({pt1})
  output$plotgraph2 = renderPlot({pt2()})
}
)

you can also play with the numbers the figure below shows c("60%", "40%")

EDIT: It is true that @Mike Wise new answer's gives some flexibility. But splitLayout can also be used with more than two plots. Using cellWidths allows you to change the size of each individual plot. And verticalLayout() can also be used to add plots vertically (see comment section).

library(shiny)
library(ggplot2)
u<- shinyUI(fluidPage(
  titlePanel("title panel"),

  sidebarLayout(position = "left",
                sidebarPanel("sidebar panel",
                             checkboxInput("do2", "Make 2 plots", value = T)
                ),
                mainPanel("main panel",
                          fluidRow(
                            splitLayout(style = "border: 1px solid silver:", cellWidths = c(300,200,100), 
                            plotOutput("plotgraph1"), 
                            plotOutput("plotgraph2"),
                            plotOutput("plotgraph3")
                            )
                          )
                )
  )
)
)
s <- shinyServer(function(input, output){
  set.seed(1234)
  pt1 <- qplot(rnorm(500),fill=I("red"),binwidth=0.2,title="plotgraph1")
  pt3 <- qplot(rnorm(600),fill=I("blue"),binwidth=0.2,title="plotgraph3")
  pt2 <- reactive({
    input$do2
    if (input$do2){
      return(qplot(rnorm(500),fill=I("blue"),binwidth=0.2,title="plotgraph2"))
    } else {
      return(NULL)
    }
  })
  output$plotgraph1 = renderPlot({pt1})
  output$plotgraph2 = renderPlot({pt2()})
  output$plotgraph3 = renderPlot({pt3}
  )
})

shinyApp(u,s)