Q:

How to reorganize the page using "div" and "css"? 

I'm not sure how to move search field with blue button in red rectangle to be right of "Поиск мероприятий:" label? 
Here's the code:
<style>
    #header {
        border: 4px solid red;
        background-color:white;
        color:black;
        text-align:left;
    }
    #nav {
        border: 4px solid green;
        line-height:20px;
        height:420px;
        width:250px;
        float:left;
        padding:5px;          
    }
    #section1 {
        border: 4px solid blue;
        width:700px;
        float:left;
        padding:5px;         
    }
    #section2 {
        border: 4px solid brown;
        background-color: gainsboro;
        width:700px;
        float:right;
        padding:5px;         
    }
</style>

<div id="header">
    <h2>Поиск мероприятий:</h2>
    <div id="searchKeyword" class="input-group">
        <i class="glyphicon glyphicon-search input-group-addon"></i>
        <input type="text" class="form-control" placeholder="Часть названия мероприятия" ng-model="actionsQuery.searchText" on-enter="queryResult()">
        <! --<span class="input-group-btn" ng-hide="!actionsQuery.searchText">-->
        <span class="input-group-btn">
            <button class="btn btn-primary" type="button" ng-click="queryResult()">Найти мероприятия</button>
        </span>
    </div>
</div>

<div id="nav">
    ...
</div>

<div id="section1">
    ...
</div>

<div id="section2">
    ...
</div>

A:

Use CSS display: inline-block
#header h2, #searchKeyword{
       display:inline-block;
    }  

#header {
        border: 4px solid red;
        background-color:white;
        color:black;
        text-align:left;
    }
    #header h2, #searchKeyword{
       display:inline-block;
    }
    #nav {
        border: 4px solid green;
        line-height:20px;
        height:420px;
        width:250px;
        float:left;
        padding:5px;          
    }
    #section1 {
        border: 4px solid blue;
        width:700px;
        float:left;
        padding:5px;         
    }
    #section2 {
        border: 4px solid brown;
        background-color: gainsboro;
        width:700px;
        float:right;
        padding:5px;         
    }
<div id="header">
    <h2>Поиск мероприятий:</h2>
    <div id="searchKeyword" class="input-group">
        <i class="glyphicon glyphicon-search input-group-addon"></i>
        <input type="text" class="form-control" placeholder="Часть названия мероприятия" ng-model="actionsQuery.searchText" on-enter="queryResult()">
        <! --<span class="input-group-btn" ng-hide="!actionsQuery.searchText">-->
        <span class="input-group-btn">
            <button class="btn btn-primary" type="button" ng-click="queryResult()">Найти мероприятия</button>
        </span>
    </div>
</div>

<div id="nav">
    ...
</div>

<div id="section1">
    ...
</div>

<div id="section2">
    ...
</div>

Or Use float - CSS 
#header h2, #searchKeyword{
   float:left;
}

#header {
    border: 4px solid red;
    background-color:white;
    color:black;
    text-align:left;
    overflow: auto;
}
#header h2, #searchKeyword{
   float:left;
}
#searchKeyword{
   line-height:64px;
}
#nav {
    border: 4px solid green;
    line-height:20px;
    height:420px;
    width:250px;
    float:left;
    padding:5px;          
}
#section1 {
    border: 4px solid blue;
    width:700px;
    float:left;
    padding:5px;         
}
#section2 {
    border: 4px solid brown;
    background-color: gainsboro;
    width:700px;
    float:right;
    padding:5px;         
}
<div id="header">
    <h2>Поиск мероприятий:</h2>
    <div id="searchKeyword" class="input-group">
        <i class="glyphicon glyphicon-search input-group-addon"></i>
        <input type="text" class="form-control" placeholder="Часть названия мероприятия" ng-model="actionsQuery.searchText" on-enter="queryResult()">
        <! --<span class="input-group-btn" ng-hide="!actionsQuery.searchText">-->
        <span class="input-group-btn">
            <button class="btn btn-primary" type="button" ng-click="queryResult()">Найти мероприятия</button>
        </span>
    </div>
</div>

<div id="nav">
    ...
</div>

<div id="section1">
    ...
</div>

<div id="section2">
    ...
</div>

