The problem is that you need to have single quotes around the value for Type. Using string concatenation is frowned upon when working with SQL because of SQL injection and other issues. You should get in the habit of using parameterized queries.
string cmd = "INSERT INTO Log(Type, Rego) VALUES('" + typeBox.Text + "'," + regoBox.Text + ")";
Parameterized Query Version.
SqlCeConnection conn = new SqlCeConnection(connStr);
conn.Open();
string cmd = "INSERT INTO Log(Type, Rego) VALUES(@Type, @Rego)";
SqlCeCommand writeToDB = new SqlCeCommand(cmd, conn);
writeToDB.Parameters.Add(new SqlCeParameter("@Type", typeBox.Text);
writeToDB.Parameters.Add(new SqlCeParameter("@Rego", regoBox.Text);
writeToDB.ExecuteNonQuery();
conn.Close();