Hi sonali verma,

every thing is fine in your code, here the problem is in your contact query, you are trying to query list of contacts having name as its account id, change the query as 

List<Contact> conList=[select LastName,email from Contact where AccountId=:con.AccountID];

and use 

<apex:column value="{!c.LastName}"/>

Instead of  using 

<apex:outputtext value="{!c.LastName}"/>

Please change your code as below, it will work.

<apex:page standardController="contact"
           extensions="SelectAccountThroughLookup">
    
    <apex:form >
        <apex:pageBlock title="Select Account Name from Lookup">
            <apex:outputLabel >Select Account Name</apex:outputLabel>
            
            <apex:inputField id="field1"
                             value="{!Contact.AccountID}">
                
                <apex:actionSupport event="onchange"
                                    id="id1" 
                                    reRender="check"/>
                
            </apex:inputField>
        </apex:pageBlock>
        
        <apex:outputPanel id="check">
            <apex:pageBlock title="Contact Details">
                <apex:pageBlockTable value="{!showContactName}"
                                     var="c">
                    <apex:column value="{!c.LastName}"/>
                    <apex:outputtext value="{!c.LastName}"/>
                    <apex:column value="{!c.email}"/>
                </apex:pageBlockTable>
                
            </apex:pageBlock>
        </apex:outputPanel>
    </apex:form>           
</apex:page>


public class SelectAccountThroughLookup
{
    //I need to select account name from contact lookup. so I need variable of type contact
    private Contact con;
    
    public boolean check{get;set;}
    
    public SelectAccountThroughLookup(ApexPages.StandardController stdcontroller) 
    {
      this.con=(Contact)stdcontroller.getRecord();
    }
    
    public SelectAccountThroughLookup()
    {
      //check=false;
    }
    
    public List<Contact> getShowContactName()
    {
        system.debug('con'+con);
      List<Contact> conList=[select LastName,email from Contact where AccountId=:con.AccountID];
        system.debug('listOf contacts'+conList);
      return conList;
    }
 
}

Please let me know, if it works for you.

Best Regards,
Dayakar.D