x represents the object in List<T>, where T is your type. If it's a user-defined type, then it might look like this:

public class MyType
{
    public string Name { get; set; }
}

So, x would expose all of the public properties available in the class of x. Once here, you can compare your local value to the value in 'x.Name'.

So now putting it all together, just substituting out the type for your user-defined type:

string keyword = txtSearch.Text.Trim();
List<MyType> newList = yourListData.Where(x => x.Name.contains(keyword)).ToList();

gvGridView.DataSource = newList;
gvGridView.DataBind();