Shalom and thank you for your question! As far as I understand your question, you wish to clarify if sacrifices are brought for intentional sins, or for unintentional sins, or both. In fact (in the sanctuary during the period that the Jewish people were in the desert and during the eras of the first and the second Temples,) sacrifices were brought in both cases, but there are many different kinds of sacrifices for different purposes, and we will not go into the details here, but speak about the deeper meaning of the sacrifices.
The reason that a sacrifice needs to be brought also for an unintentional sin, is that a situation was brought about which was contrary to G-d’s will, as expressed in the teaching and instruction of the Torah. The Torah is the Divine will and wisdom, and the Sages describe it as the blueprint by which G-d created and recreates the world. When something has taken place that is contrary to this, it has cosmic repercussions. Thus, atonement needs to be made. You can read atonement as ‘at-one-ment’. When G-d’s instructions of what to do in order to rectify what went wrong are carried out, the spiritual ‘hole in the ozone layer’ is repaired. The negative effect that the sin had is rectified.
The sins involving monetary issues between people, which are referred to in Leviticus Chapter 5 which you ask about, usually require direct restitution of the money to the party that was wronged. This applies most definitely in our times, and is not dependent on having the Temple rebuilt.
There are two basic categories of sin, which include sins between man and G-d, and sins between man and man. In general when a person wrongs another in any way, he/she must ask forgiveness (in certain circumstances where it may cause embarrassment to the person who was wronged this may be waived,) of the wronged party, and make monetary restitution if necessary.
In the case of the type of sin that does require sacrifices, in the times when that was relevant, and in the future when the Holy Temple will be rebuilt, the sacrifice itself is NOT enough to effect atonement. As I mentioned above, when G-d’s instructions are carried out, this rectifies the negative effects of the sin, but it involves many aspects. When sacrifices were brought, the Levites would be singing spiritually arousing songs and playing instruments, the atmosphere was thus charged with spirituality. Both the person bringing the sacrifice and the Cohen (priest,) offering it up to the L-rd had to be focused on atonement for the specific sin involved, (or on thanksgiving if it was a thanksgiving sacrifice.) and if they were not focused on the correct matter, the sacrifice was invalid!
In our day prayer is considered a replacement for bringing sacrifices, and as above-mentioned, it requires sincerity and focus. It is of utmost importance that we rectify our misdeeds by a process of acknowledging the wrong that was done, regretting it and resolving to correct it, making restitution as mentioned above, and following a better path from then on.
