When we speak of the mood of a piece, we're referring to the reader's emotions that the writer of the piece evokes through word choices and even literary devices ("Mood"). Any word used to describe an emotion can be used to describe the mood of a piece, like "peaceful," "contemplative," "frustrated," "disappointed," and even "nostalgic" ("Mood Words").
The mood Henry Wadsworth Longfellow creates in his poem "My Lost Youth" can actually be debated. On the one hand, many of the images Longfellow conveys in the poem are upbeat. While it is clear the poem's speaker is reflecting on his past, most of those past reflections are very positive, as can be seen in the line describing the "pleasant streets of that dear old town" (I.4). Hence, some interpreters will characterize the mood as being optimistic because the speaker is optimistically reflecting back on his past.
On the other hand, there certainly is also some negative diction, particularly used with respect to the poem's refrain. Specifically, at different points, the poem's refrain is introduced with phrases like "burden of that old song" and "fitful song" (2.6, 7.6). At one point, the refrain is even introduced with the phrase "fatal song," as we see in the following lines:
And the words of that fatal song
Come over me like a chill:
"A boy's will is the wind's will,
And the thoughts of youth are long, long thoughts." (8.6-9)
Essentially, the refrain is describing the will, or the desires, of a young boy as being restless, just like the wind. Just as the wind blows every which way, changing directions without a moment's notice, so do the desires of a boy change frequently. Hence, in using negative diction alongside the refrain, Longfellow may be capturing the speaker's regret that he was not more grounded growing up as a youth, that he did not have a steady direction as a youth. Hence, it can also be said that the mood is regretful.
All in all, the poem seems to contain a very complex mood, a bitter-sweet mood. The mood is bitter as he reflects on a few negative memories and his desire to have been more grounded; it is sweet as he contentedly reflects on his past childhood.