Search for value in DataGridView in a column

Why you are using row.Cells[row.Index]. You need to specify index of column you want to search (Problem #2). For example, you need to change row.Cells[row.Index] to row.Cells[2] where 2 is index of your column:

private void btnSearch_Click(object sender, EventArgs e)
{
    string searchValue = textBox1.Text;

    dgvProjects.SelectionMode = DataGridViewSelectionMode.FullRowSelect;
    try
    {
        foreach (DataGridViewRow row in dgvProjects.Rows)
        {
            if (row.Cells[2].Value.ToString().Equals(searchValue))
            {
                row.Selected = true;
                break;
            }
        }
    }
    catch (Exception exc)
    {
        MessageBox.Show(exc.Message);
    }
}