Reflexives?
 

I seeing these sentences and no explanation as to the differences.

Me pongo el sombrero
Me puse el vestido

Nos ponemos el abrigo
Nos ponimos la bufanda

I know that the first one is present tense and the second is past, but when would you ever use the first sentence as a present tense? I can only think that it might be used as a sentence to explain what you do, so you might say on a sunny day "Me pongo el sombrero" but the book still has no explanation and I am left guessing.


Late edit: Now that I think of it, I am thinking that it has to do with me not knowing any past tense words for put. For instance I can say "I get up" or "I got up".

We use them to speak about actions that we do ourselves. Reflexive pronouns are used with reflexive verbs. And they can be used in all tenses, depending on what you want to express, either, present, past o future.
Hoy me he levantado a las 8. Levantarse is the reflexive verb.
Me pongo el abrigo cuando hace frío. ( ponerse) habitual action
Me afeitaré mañana antes de ir al trabajo. ( future action)
Ayer me ví en un aprieto. ( past action)
Te importaría echar de comer a los gatos. ( conditional- a request)
Me estoy enfadando contigo. (Present continuous) Now
It´s basically like in English, though in Spanish they are more widely used.

Nos ponemos la bufanda

or

Nos pusimos la bufanda

Well spotted that about "put". I got the feeling I'm leaving out any temporal information when I use put or cut, or when I write read.

You're right "nos ponemos la bufanda" sounds atypical and somehow purposeless, but you may hear such kind of phrases in real life when accounting for habits or repetitive actions, as Robin already told, or as instructions aimed to kids (as the impersonal "ponerse la bufanda" is not clearly understood by a three year old kid), or as a softened command, generally aimed to children too. Of course, in the last two instances it's more common to hear "nos ponemos las bufandas", for identical reasons.

Without any further context, I assume the book is simply giving examples of what reflexive verbs are and how they are constructed in present and past tenses.
When the subject in a sentence performs an action on itself we use reflexive verbs. Since the most common are those talking about daily habits like washing oneself (lavarse), getting dressed (vestirse), comb one's hair (peinarse), etc., your book is using an alternative verb to "vestirse", which is "ponerse [ropa]" (to put [clothes] on).



Side note: Robin gave you a fine list of extra examples.
I would just put aside "Te importaría echar de comer a los gatos", because "te" here is an indirect object pronoun instead of a reflexive one. :)